Theorem foeq3 5104

Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
foeq3	|- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )

Proof of Theorem foeq3

Step	Hyp	Ref	Expression
1		eqeq2 2049	. . 3 |- ( A = B -> ( ran F = A <-> ran F = B ) )
2	1	anbi2d 437	. 2 |- ( A = B -> ( ( F Fn C /\ ran F = A ) <-> ( F Fn C /\ ran F = B ) ) )
3		df-fo 4908	. 2 |- ( F : C -onto-> A <-> ( F Fn C /\ ran F = A ) )
4		df-fo 4908	. 2 |- ( F : C -onto-> B <-> ( F Fn C /\ ran F = B ) )
5	2, 3, 4	3bitr4g 212	1 |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   ran crn 4346   Fn wfn 4897   -onto->wfo 4900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-fo 4908

This theorem is referenced by:  f1oeq3  5119  foeq123d  5122  resdif  5148  ffoss  5158