Theorem foeq2 5497

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 5364	. . 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 5278	. 2 |- ( F : A -onto-> C <-> ( F Fn A /\ ran F = C ) )
4		df-fo 5278	. 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 4677   Fn wfn 5267   -onto->wfo 5270

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 1470  ax-gen 1472  ax-4 1533  ax-17 1549  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-fn 5275  df-fo 5278

This theorem is referenced by:  f1oeq2  5513  foeq123d  5517  tposfo  6359  ctssdclemr  7216  enumct  7219  exmidfodomrlemr  7312  exmidfodomrlemrALT  7313  ctinf  12834  ctiunct  12844  ssomct  12849  subctctexmid  15974  domomsubct  15975