let A, B be non empty AltGraph ; :: thesis: for F being BimapStr over A,B st F is Covariant & F is one-to-one holds

for a, b being Object of A st F . a = F . b holds

a = b

let F be BimapStr over A,B; :: thesis: ( F is Covariant & F is one-to-one implies for a, b being Object of A st F . a = F . b holds

a = b )

given f being Function of the carrier of A, the carrier of B such that A1: the ObjectMap of F = [:f,f:] ; :: according to FUNCTOR0:def 1,FUNCTOR0:def 12 :: thesis: ( not F is one-to-one or for a, b being Object of A st F . a = F . b holds

a = b )

assume the ObjectMap of F is V7() ; :: according to FUNCTOR0:def 6 :: thesis: for a, b being Object of A st F . a = F . b holds

a = b

then A2: f is one-to-one by A1, FUNCTOR0:7;

let a, b be Object of A; :: thesis: ( F . a = F . b implies a = b )

assume A3: F . a = F . b ; :: thesis: a = b

A4: ( [(f . a),(f . a)] `1 = f . a & [(f . b),(f . b)] `1 = f . b ) ;

( the ObjectMap of F . (a,a) = [(f . a),(f . a)] & the ObjectMap of F . (b,b) = [(f . b),(f . b)] ) by A1, FUNCT_3:75;

hence a = b by A2, A3, A4, FUNCT_2:19; :: thesis: verum

for a, b being Object of A st F . a = F . b holds

a = b

let F be BimapStr over A,B; :: thesis: ( F is Covariant & F is one-to-one implies for a, b being Object of A st F . a = F . b holds

a = b )

given f being Function of the carrier of A, the carrier of B such that A1: the ObjectMap of F = [:f,f:] ; :: according to FUNCTOR0:def 1,FUNCTOR0:def 12 :: thesis: ( not F is one-to-one or for a, b being Object of A st F . a = F . b holds

a = b )

assume the ObjectMap of F is V7() ; :: according to FUNCTOR0:def 6 :: thesis: for a, b being Object of A st F . a = F . b holds

a = b

then A2: f is one-to-one by A1, FUNCTOR0:7;

let a, b be Object of A; :: thesis: ( F . a = F . b implies a = b )

assume A3: F . a = F . b ; :: thesis: a = b

A4: ( [(f . a),(f . a)] `1 = f . a & [(f . b),(f . b)] `1 = f . b ) ;

( the ObjectMap of F . (a,a) = [(f . a),(f . a)] & the ObjectMap of F . (b,b) = [(f . b),(f . b)] ) by A1, FUNCT_3:75;

hence a = b by A2, A3, A4, FUNCT_2:19; :: thesis: verum