Theorem foeq3 5548

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 2239	. . 3 |- ( A = B -> ( ran F = A <-> ran F = B ) )
2	1	anbi2d 464	. 2 |- ( A = B -> ( ( F Fn C /\ ran F = A ) <-> ( F Fn C /\ ran F = B ) ) )
3		df-fo 5324	. 2 |- ( F : C -onto-> A <-> ( F Fn C /\ ran F = A ) )
4		df-fo 5324	. 2 |- ( F : C -onto-> B <-> ( F Fn C /\ ran F = B ) )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( F : C -onto-> A <-> F : C -onto-> B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   ran crn 4720   Fn wfn 5313   -onto->wfo 5316

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-fo 5324

This theorem is referenced by:  fimadmfo  5559  f1oeq3  5564  foeq123d  5567  resdif  5596  ffoss  5606  fifo  7155  enumct  7290  ctssexmid  7325  exmidfodomrlemr  7388  exmidfodomrlemrALT  7389  qnnen  13010  ctiunctal  13020  unct  13021  quslem  13365  znzrhfo  14620  gausslemma2dlem1f1o  15747  subctctexmid  16392