Theorem foeq2 5615

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 5473	. . 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 5386	. 2 |- ( F : A -onto-> C <-> ( F Fn A /\ ran F = C ) )
4		df-fo 5386	. 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 4777   Fn wfn 5375   -onto->wfo 5378

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 5383  df-fo 5386

This theorem is used by:  f1oeq2  5631  foeq123d  5635  tposfo  6546  ctssdclemr  7458  enumct  7461  exmidfodomrlemr  7560  exmidfodomrlemrALT  7561  ctinf  13379  ctiunct  13389  ssomct  13394  subctctexmid  17235  domomsubct  17236