Theorem foeq3 5182

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 2094	. . 3 |- ( A = B -> ( ran F = A <-> ran F = B ) )
2	1	anbi2d 452	. 2 |- ( A = B -> ( ( F Fn C /\ ran F = A ) <-> ( F Fn C /\ ran F = B ) ) )
3		df-fo 4978	. 2 |- ( F : C -onto-> A <-> ( F Fn C /\ ran F = A ) )
4		df-fo 4978	. 2 |- ( F : C -onto-> B <-> ( F Fn C /\ ran F = B ) )
5	2, 3, 4	3bitr4g 221	1 |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1287   ran crn 4405   Fn wfn 4967   -onto->wfo 4970

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-4 1443  ax-17 1462  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-fo 4978

This theorem is referenced by:  f1oeq3  5197  foeq123d  5200  resdif  5226  ffoss  5236  exmidfodomrlemr  6749  exmidfodomrlemrALT  6750