Theorem foeq2 5612

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

Assertion

Ref	Expression
foeq2	|- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )

Proof of Theorem foeq2

Step	Hyp	Ref	Expression
1		fneq2 5470	. . 3 |- ( A = B -> ( F Fn A <-> F Fn B ) )
2	1	anbi1d 469	. 2 |- ( A = B -> ( ( F Fn A /\ ran F = C ) <-> ( F Fn B /\ ran F = C ) ) )
3		df-fo 5383	. 2 |- ( F : A -onto-> C <-> ( F Fn A /\ ran F = C ) )
4		df-fo 5383	. 2 |- ( F : B -onto-> C <-> ( F Fn B /\ ran F = C ) )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   ran crn 4775   Fn wfn 5372   -onto->wfo 5375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-fn 5380  df-fo 5383

This theorem is used by:  f1oeq2  5628  foeq123d  5632  tposfo  6542  ctssdclemr  7452  enumct  7455  exmidfodomrlemr  7554  exmidfodomrlemrALT  7555  ctinf  13321  ctiunct  13331  ssomct  13336  subctctexmid  17030  domomsubct  17031