Theorem foeq2 5406

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 5276	. . 3 |- ( A = B -> ( F Fn A <-> F Fn B ) )
2	1	anbi1d 461	. 2 |- ( A = B -> ( ( F Fn A /\ ran F = C ) <-> ( F Fn B /\ ran F = C ) ) )
3		df-fo 5193	. 2 |- ( F : A -onto-> C <-> ( F Fn A /\ ran F = C ) )
4		df-fo 5193	. 2 |- ( F : B -onto-> C <-> ( F Fn B /\ ran F = C ) )
5	2, 3, 4	3bitr4g 222	1 |- ( A = B -> ( F : A -onto-> C <-> F : B -onto-> C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   ran crn 4604   Fn wfn 5182   -onto->wfo 5185

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-fn 5190  df-fo 5193

This theorem is referenced by:  f1oeq2  5421  foeq123d  5425  tposfo  6235  ctssdclemr  7073  enumct  7076  exmidfodomrlemr  7154  exmidfodomrlemrALT  7155  ctinf  12359  ctiunct  12369  ssomct  12374  subctctexmid  13841