Theorem foeq3 5408

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 2175	. . 3 |- ( A = B -> ( ran F = A <-> ran F = B ) )
2	1	anbi2d 460	. 2 |- ( A = B -> ( ( F Fn C /\ ran F = A ) <-> ( F Fn C /\ ran F = B ) ) )
3		df-fo 5194	. 2 |- ( F : C -onto-> A <-> ( F Fn C /\ ran F = A ) )
4		df-fo 5194	. 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 1343   ran crn 4605   Fn wfn 5183   -onto->wfo 5186

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 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-fo 5194

This theorem is referenced by:  f1oeq3  5423  foeq123d  5426  resdif  5454  ffoss  5464  fifo  6945  enumct  7080  ctssexmid  7114  exmidfodomrlemr  7158  exmidfodomrlemrALT  7159  qnnen  12364  ctiunctal  12374  unct  12375  subctctexmid  13881