Theorem foeq2 5382

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 5252	. . 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 5169	. 2 |- ( F : A -onto-> C <-> ( F Fn A /\ ran F = C ) )
4		df-fo 5169	. 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 4580   Fn wfn 5158   -onto->wfo 5161

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 1487  ax-17 1503  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-cleq 2147  df-fn 5166  df-fo 5169

This theorem is referenced by:  f1oeq2  5397  foeq123d  5401  tposfo  6208  ctssdclemr  7042  enumct  7045  exmidfodomrlemr  7116  exmidfodomrlemrALT  7117  ctinf  12110  ctiunct  12120  subctctexmid  13512