let A, I, y be non empty set ; :: thesis: for F being Function of A,I holds { z where z is Element of A : F . z in y } = F " y

let F be Function of A,I; :: thesis: { z where z is Element of A : F . z in y } = F " y

for x being object holds

( x in { z where z is Element of A : F . z in y } iff x in F " y )

let F be Function of A,I; :: thesis: { z where z is Element of A : F . z in y } = F " y

for x being object holds

( x in { z where z is Element of A : F . z in y } iff x in F " y )

proof

hence
{ z where z is Element of A : F . z in y } = F " y
by TARSKI:2; :: thesis: verum
let x be object ; :: thesis: ( x in { z where z is Element of A : F . z in y } iff x in F " y )

then ( x in dom F & F . x in y ) by FUNCT_1:def 7;

hence x in { z where z is Element of A : F . z in y } ; :: thesis: verum

end;hereby :: thesis: ( x in F " y implies x in { z where z is Element of A : F . z in y } )

assume
x in F " y
; :: thesis: x in { z where z is Element of A : F . z in y } assume
x in { z where z is Element of A : F . z in y }
; :: thesis: x in F " y

then consider z being Element of A such that

A1: ( x = z & F . z in y ) ;

z in A ;

then z in dom F by FUNCT_2:def 1;

hence x in F " y by FUNCT_1:def 7, A1; :: thesis: verum

end;then consider z being Element of A such that

A1: ( x = z & F . z in y ) ;

z in A ;

then z in dom F by FUNCT_2:def 1;

hence x in F " y by FUNCT_1:def 7, A1; :: thesis: verum

then ( x in dom F & F . x in y ) by FUNCT_1:def 7;

hence x in { z where z is Element of A : F . z in y } ; :: thesis: verum