Theorem foeq2 5350

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 5220	. . 3 |- ( A = B -> ( F Fn A <-> F Fn B ) )
2	1	anbi1d 461	. 2 |- ( A = B -> ( ( F Fn A /\ ran F = C ) <-> ( F Fn B /\ ran F = C ) ) )
3		df-fo 5137	. 2 |- ( F : A -onto-> C <-> ( F Fn A /\ ran F = C ) )
4		df-fo 5137	. 2 |- ( F : B -onto-> C <-> ( F Fn B /\ ran F = C ) )
5	2, 3, 4	3bitr4g 222	1 |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   ran crn 4548   Fn wfn 5126   -onto->wfo 5129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-4 1488  ax-17 1507  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-fn 5134  df-fo 5137

This theorem is referenced by:  f1oeq2  5365  foeq123d  5369  tposfo  6176  ctssdclemr  7005  enumct  7008  exmidfodomrlemr  7075  exmidfodomrlemrALT  7076  ctinf  11979  ctiunct  11989  subctctexmid  13369