Theorem foeq3 5418

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

Assertion

Ref	Expression
foeq3	|- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )

Proof of Theorem foeq3

Step	Hyp	Ref	Expression
1		eqeq2 2180	. . 3 |- ( A = B -> ( ran F = A <-> ran F = B ) )
2	1	anbi2d 461	. 2 |- ( A = B -> ( ( F Fn C /\ ran F = A ) <-> ( F Fn C /\ ran F = B ) ) )
3		df-fo 5204	. 2 |- ( F : C -onto-> A <-> ( F Fn C /\ ran F = A ) )
4		df-fo 5204	. 2 |- ( F : C -onto-> B <-> ( F Fn C /\ ran F = B ) )
5	2, 3, 4	3bitr4g 222	1 |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   ran crn 4612   Fn wfn 5193   -onto->wfo 5196

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 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-fo 5204

This theorem is referenced by:  f1oeq3  5433  foeq123d  5436  resdif  5464  ffoss  5474  fifo  6957  enumct  7092  ctssexmid  7126  exmidfodomrlemr  7179  exmidfodomrlemrALT  7180  qnnen  12386  ctiunctal  12396  unct  12397  subctctexmid  14034