Theorem foeq3 5343

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 2149	. . 3 |- ( A = B -> ( ran F = A <-> ran F = B ) )
2	1	anbi2d 459	. 2 |- ( A = B -> ( ( F Fn C /\ ran F = A ) <-> ( F Fn C /\ ran F = B ) ) )
3		df-fo 5129	. 2 |- ( F : C -onto-> A <-> ( F Fn C /\ ran F = A ) )
4		df-fo 5129	. 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 1331   ran crn 4540   Fn wfn 5118   -onto->wfo 5121

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 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-fo 5129

This theorem is referenced by:  f1oeq3  5358  foeq123d  5361  resdif  5389  ffoss  5399  fifo  6868  enumct  7000  ctssexmid  7024  exmidfodomrlemr  7058  exmidfodomrlemrALT  7059  qnnen  11944  unct  11954  subctctexmid  13196