Theorem foeq2 5517

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

Assertion

Ref	Expression
foeq2	|- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )

Proof of Theorem foeq2

Step	Hyp	Ref	Expression
1		fneq2 5382	. . 3 |- ( A = B -> ( F Fn A <-> F Fn B ) )
2	1	anbi1d 465	. 2 |- ( A = B -> ( ( F Fn A /\ ran F = C ) <-> ( F Fn B /\ ran F = C ) ) )
3		df-fo 5296	. 2 |- ( F : A -onto-> C <-> ( F Fn A /\ ran F = C ) )
4		df-fo 5296	. 2 |- ( F : B -onto-> C <-> ( F Fn B /\ ran F = C ) )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   ran crn 4694   Fn wfn 5285   -onto->wfo 5288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-cleq 2200  df-fn 5293  df-fo 5296

This theorem is referenced by:  f1oeq2  5533  foeq123d  5537  tposfo  6380  ctssdclemr  7240  enumct  7243  exmidfodomrlemr  7341  exmidfodomrlemrALT  7342  ctinf  12916  ctiunct  12926  ssomct  12931  subctctexmid  16139  domomsubct  16140