Theorem foeq2 5556

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 5419	. . 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 5332	. 2 |- ( F : A -onto-> C <-> ( F Fn A /\ ran F = C ) )
4		df-fo 5332	. 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 1397   ran crn 4726   Fn wfn 5321   -onto->wfo 5324

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-fn 5329  df-fo 5332

This theorem is referenced by:  f1oeq2  5572  foeq123d  5576  tposfo  6436  ctssdclemr  7310  enumct  7313  exmidfodomrlemr  7412  exmidfodomrlemrALT  7413  ctinf  13050  ctiunct  13060  ssomct  13065  subctctexmid  16601  domomsubct  16602