Theorem foeq3 5458

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 2199	. . 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 5244	. 2 |- ( F : C -onto-> A <-> ( F Fn C /\ ran F = A ) )
4		df-fo 5244	. 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 1364   ran crn 4648   Fn wfn 5233   -onto->wfo 5236

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-fo 5244

This theorem is referenced by:  f1oeq3  5473  foeq123d  5476  resdif  5505  ffoss  5515  fifo  7013  enumct  7148  ctssexmid  7183  exmidfodomrlemr  7236  exmidfodomrlemrALT  7237  qnnen  12493  ctiunctal  12503  unct  12504  quslem  12812  subctctexmid  15237