Volume 1, 1989

University of Bialystok

Copyright (c) 1989 Association of Mizar Users

### The abstract of the Mizar article:

### Basic Functions and Operations on Functions

**by****Czeslaw Bylinski**- Received May 9, 1989
- MML identifier: FUNCT_3

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_1, RELAT_1, BOOLE, TARSKI, PARTFUN1, FUNCT_2, FUNCT_3; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2; constructors TARSKI, FUNCT_2, XBOOLE_0; clusters FUNCT_1, RELSET_1, SUBSET_1, XBOOLE_0, ZFMISC_1; requirements SUBSET, BOOLE; begin reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set; reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set; reserve C,C1,C2,D,D1,D2 for non empty set; :: Additional propositions about functions theorem :: FUNCT_3:1 A c= Y implies id A = (id Y)|A; theorem :: FUNCT_3:2 for f,g being Function st X c= dom(g*f) holds f.:X c= dom g; theorem :: FUNCT_3:3 for f,g being Function st X c= dom f & f.:X c= dom g holds X c= dom(g*f); theorem :: FUNCT_3:4 for f,g being Function st Y c= rng(g*f) & g is one-to-one holds g"Y c= rng f; theorem :: FUNCT_3:5 for f,g being Function st Y c= rng g & g"Y c= rng f holds Y c= rng(g*f); scheme FuncEx_3{A()->set,B()-> set,P[set,set,set]}: ex f being Function st dom f = [:A(),B():] & for x,y st x in A() & y in B() holds P[x,y,f.[x,y]] provided for x,y,z1,z2 st x in A() & y in B() & P[x,y,z1] & P[x,y,z2] holds z1 = z2 and for x,y st x in A() & y in B() ex z st P[x,y,z]; scheme Lambda_3{A()->set,B()->set,F(set,set)->set}: ex f being Function st dom f = [:A(),B():] & for x,y st x in A() & y in B() holds f.[x,y] = F(x,y); theorem :: FUNCT_3:6 for f,g being Function st dom f = [:X,Y:] & dom g = [:X,Y:] & for x,y st x in X & y in Y holds f.[x,y] = g.[x,y] holds f = g; :: Function indicated by the image under a function definition let f be Function; func .:f -> Function means :: FUNCT_3:def 1 dom it = bool dom f & for X st X c= dom f holds it.X = f.:X; end; canceled; theorem :: FUNCT_3:8 for f being Function st X in dom(.:f) holds (.:f).X = f.:X; theorem :: FUNCT_3:9 for f being Function holds (.:f).{} = {}; theorem :: FUNCT_3:10 for f being Function holds rng(.:f) c= bool rng f; canceled; theorem :: FUNCT_3:12 for f being Function holds (.:f).:A c= bool rng f; theorem :: FUNCT_3:13 for f being Function holds (.:f)"B c= bool dom f; theorem :: FUNCT_3:14 for f being Function of X,D holds (.:f)"B c= bool X; theorem :: FUNCT_3:15 for f being Function holds union((.:f).:A) c= f.:(union A); theorem :: FUNCT_3:16 for f being Function st A c= bool dom f holds f.:(union A) = union((.:f).:A) ; theorem :: FUNCT_3:17 for f being Function of X,D st A c= bool X holds f.:(union A) = union((.:f).:A); theorem :: FUNCT_3:18 for f being Function holds union((.:f)"B) c= f"(union B); theorem :: FUNCT_3:19 for f being Function st B c= bool rng f holds f"(union B) = union((.:f)"B) ; theorem :: FUNCT_3:20 for f,g being Function holds .:(g*f) = .:g*.:f; theorem :: FUNCT_3:21 for f being Function holds .:f is Function of bool dom f, bool rng f; theorem :: FUNCT_3:22 for f being Function of X,Y st Y = {} implies X = {} holds .:f is Function of bool X, bool Y; definition let X,D; let f be Function of X,D; redefine func .:f -> Function of bool X, bool D; end; :: Function indicated by the inverse image under a function definition let f be Function; func "f -> Function means :: FUNCT_3:def 2 dom it = bool rng f & for Y st Y c= rng f holds it.Y = f"Y; end; canceled; theorem :: FUNCT_3:24 for f being Function st Y in dom("f) holds ("f).Y = f"Y; theorem :: FUNCT_3:25 for f being Function holds rng("f) c= bool dom f; canceled; theorem :: FUNCT_3:27 for f being Function holds ("f).:B c= bool dom f; theorem :: FUNCT_3:28 for f being Function holds ("f)"A c= bool rng f; theorem :: FUNCT_3:29 for f being Function holds union(("f).:B) c= f"(union B); theorem :: FUNCT_3:30 for f being Function st B c= bool rng f holds union(("f).:B) = f"(union B) ; theorem :: FUNCT_3:31 for f being Function holds union(("f)"A) c= f.:(union A); theorem :: FUNCT_3:32 for f being Function st A c= bool dom f & f is one-to-one holds union(("f)"A) = f.:(union A); theorem :: FUNCT_3:33 for f being Function holds ("f).:B c= (.:f)"B; theorem :: FUNCT_3:34 for f being Function st f is one-to-one holds ("f).:B = (.:f)"B; theorem :: FUNCT_3:35 for f being Function,A be set st A c= bool dom f holds ("f)"A c= (.:f).:A; theorem :: FUNCT_3:36 for f being Function,A be set st f is one-to-one holds (.:f).:A c= ("f)"A; theorem :: FUNCT_3:37 for f being Function,A be set st f is one-to-one & A c= bool dom f holds ("f)"A = (.:f).:A; theorem :: FUNCT_3:38 for f,g being Function st g is one-to-one holds "(g*f) = "f*"g; theorem :: FUNCT_3:39 for f being Function holds "f is Function of bool rng f, bool dom f; :: Characteristic function definition let A,X; func chi(A,X) -> Function means :: FUNCT_3:def 3 dom it = X & for x st x in X holds (x in A implies it.x = 1) & (not x in A implies it.x = 0); end; canceled 2; theorem :: FUNCT_3:42 chi(A,X).x = 1 implies x in A; theorem :: FUNCT_3:43 x in X \ A implies chi(A,X).x = 0; canceled 3; theorem :: FUNCT_3:47 A c= X & B c= X & chi(A,X) = chi(B,X) implies A = B; theorem :: FUNCT_3:48 rng chi(A,X) c= {0,1}; theorem :: FUNCT_3:49 for f being Function of X,{0,1} holds f = chi(f"{1},X); definition let A,X; redefine func chi(A,X) -> Function of X,{0,1}; end; definition let Y; let A be Subset of Y; func incl(A) -> Function of A,Y equals :: FUNCT_3:def 4 id A; end; canceled 3; theorem :: FUNCT_3:53 for A being Subset of Y holds incl A = (id Y)|A; theorem :: FUNCT_3:54 for A being Subset of Y holds dom incl A = A & rng incl A = A; theorem :: FUNCT_3:55 for A being Subset of Y st x in A holds (incl A).x = x; theorem :: FUNCT_3:56 for A being Subset of Y st x in A holds incl(A).x in Y; :: Projections definition let X,Y; func pr1(X,Y) -> Function means :: FUNCT_3:def 5 dom it = [:X,Y:] & for x,y st x in X & y in Y holds it.[x,y] = x; func pr2(X,Y) -> Function means :: FUNCT_3:def 6 dom it = [:X,Y:] & for x,y st x in X & y in Y holds it.[x,y] = y; end; canceled 2; theorem :: FUNCT_3:59 rng pr1(X,Y) c= X; theorem :: FUNCT_3:60 Y <> {} implies rng pr1(X,Y) = X; theorem :: FUNCT_3:61 rng pr2(X,Y) c= Y; theorem :: FUNCT_3:62 X <> {} implies rng pr2(X,Y) = Y; definition let X,Y; redefine func pr1(X,Y) -> Function of [:X,Y:],X; func pr2(X,Y) -> Function of [:X,Y:],Y; end; definition let X; func delta(X) -> Function means :: FUNCT_3:def 7 dom it = X & for x st x in X holds it.x = [x,x]; end; canceled 3; theorem :: FUNCT_3:66 rng delta X c= [:X,X:]; definition let X; redefine func delta(X) -> Function of X,[:X,X:]; end; :: Complex functions definition let f,g be Function; func <:f,g:> -> Function means :: FUNCT_3:def 8 dom it = dom f /\ dom g & for x st x in dom it holds it.x = [f.x,g.x]; end; canceled; theorem :: FUNCT_3:68 for f,g being Function st x in dom f /\ dom g holds <:f,g:>.x = [f.x,g.x]; theorem :: FUNCT_3:69 for f,g being Function st dom f = X & dom g = X & x in X holds <:f,g:>.x = [f.x,g.x]; theorem :: FUNCT_3:70 for f,g being Function st dom f = X & dom g = X holds dom <:f,g:> = X; theorem :: FUNCT_3:71 for f,g being Function holds rng <:f,g:> c= [:rng f,rng g:]; theorem :: FUNCT_3:72 for f,g being Function st dom f = dom g & rng f c= Y & rng g c= Z holds pr1(Y,Z)*<:f,g:> = f & pr2(Y,Z)*<:f,g:> = g; theorem :: FUNCT_3:73 <:pr1(X,Y),pr2(X,Y):> = id [:X,Y:]; theorem :: FUNCT_3:74 for f,g,h,k being Function st dom f = dom g & dom k = dom h & <:f,g:> = <:k,h:> holds f = k & g = h; theorem :: FUNCT_3:75 for f,g,h being Function holds <:f*h,g*h:> = <:f,g:>*h; theorem :: FUNCT_3:76 for f,g being Function holds <:f,g:>.:A c= [:f.:A,g.:A:]; theorem :: FUNCT_3:77 for f,g being Function holds <:f,g:>"[:B,C:] = f"B /\ g"C; theorem :: FUNCT_3:78 for f being Function of X,Y for g being Function of X,Z st (Y = {} implies X = {}) & (Z = {} implies X = {}) holds <:f,g:> is Function of X,[:Y,Z:]; definition let X,D1,D2; let f1 be Function of X,D1; let f2 be Function of X,D2; redefine func <:f1,f2:> -> Function of X,[:D1,D2:]; end; theorem :: FUNCT_3:79 for f1 being Function of C,D1 for f2 being Function of C,D2 for c being Element of C holds <:f1,f2:>.c = [f1.c,f2.c]; theorem :: FUNCT_3:80 for f being Function of X,Y for g being Function of X,Z holds rng <:f,g:> c= [:Y,Z:]; theorem :: FUNCT_3:81 for f being Function of X,Y for g being Function of X,Z st (Y = {} implies X = {}) & (Z = {} implies X = {}) holds pr1(Y,Z)*<:f,g:> = f & pr2(Y,Z)*<:f,g:> = g; theorem :: FUNCT_3:82 for f being Function of X,D1 for g being Function of X,D2 holds pr1(D1,D2)*<:f,g:> = f & pr2(D1,D2)*<:f,g:> = g; theorem :: FUNCT_3:83 for f1,f2 being Function of X,Y for g1,g2 being Function of X,Z st (Y = {} implies X = {}) & (Z = {} implies X = {}) & <:f1,g1:> = <:f2,g2:> holds f1 = f2 & g1 = g2; theorem :: FUNCT_3:84 for f1,f2 being Function of X,D1 for g1,g2 being Function of X,D2 st <:f1,g1:> = <:f2,g2:> holds f1 = f2 & g1 = g2; :: Product-functions definition let f,g be Function; func [:f,g:] -> Function means :: FUNCT_3:def 9 dom it = [:dom f, dom g:] & for x,y st x in dom f & y in dom g holds it.[x,y] = [f.x,g.y]; end; canceled; theorem :: FUNCT_3:86 for f,g being Function, x,y st [x,y] in [:dom f,dom g:] holds [:f,g:].[x,y] = [f.x,g.y]; theorem :: FUNCT_3:87 for f,g being Function holds [:f,g:] = <:f*pr1(dom f,dom g),g*pr2(dom f,dom g):>; theorem :: FUNCT_3:88 for f,g being Function holds rng [:f,g:] = [:rng f,rng g:]; theorem :: FUNCT_3:89 for f,g being Function st dom f = X & dom g = X holds <:f,g:> = [:f,g:]*(delta X); theorem :: FUNCT_3:90 [:id X, id Y:] = id [:X,Y:]; theorem :: FUNCT_3:91 for f,g,h,k being Function holds [:f,h:]*<:g,k:> = <:f*g,h*k:>; theorem :: FUNCT_3:92 for f,g,h,k being Function holds [:f,h:]*[:g,k:] = [:f*g,h*k:]; theorem :: FUNCT_3:93 for f,g being Function holds [:f,g:].:[:B,A:] = [:f.:B,g.:A:]; theorem :: FUNCT_3:94 for f,g being Function holds [:f,g:]"[:B,A:] = [:f"B,g"A:]; theorem :: FUNCT_3:95 for f being Function of X,Y for g being Function of V,Z holds [:f,g:] is Function of [:X,V:],[:Y,Z:]; definition let X1,X2,Y1,Y2; let f1 be Function of X1,Y1; let f2 be Function of X2,Y2; redefine func [:f1,f2:] -> Function of [:X1,X2:],[:Y1,Y2:]; end; theorem :: FUNCT_3:96 for f1 being Function of C1,D1 for f2 being Function of C2,D2 for c1 being Element of C1 for c2 being Element of C2 holds [:f1,f2:].[c1,c2] = [f1.c1,f2.c2]; theorem :: FUNCT_3:97 for f1 being Function of X1,Y1 for f2 being Function of X2,Y2 st (Y1 = {} implies X1 = {}) & (Y2 = {} implies X2 = {}) holds [:f1,f2:] = <:f1*pr1(X1,X2),f2*pr2(X1,X2):>; theorem :: FUNCT_3:98 for f1 being Function of X1,D1 for f2 being Function of X2,D2 holds [:f1,f2:] = <:f1*pr1(X1,X2),f2*pr2(X1,X2):>; theorem :: FUNCT_3:99 for f1 being Function of X,Y1 for f2 being Function of X,Y2 st (Y1 = {} implies X = {}) & (Y2 = {} implies X = {}) holds <:f1,f2:> = [:f1,f2:]*(delta X); theorem :: FUNCT_3:100 for f1 being Function of X,D1 for f2 being Function of X,D2 holds <:f1,f2:> = [:f1,f2:]*(delta X);

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