Theorem feq2 5492

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

Assertion

Ref	Expression
feq2	|- ( A = B -> ( F : A --> C <-> F : B --> C ) )

Proof of Theorem feq2

Step	Hyp	Ref	Expression
1		fneq2 5445	. . 3 |- ( A = B -> ( F Fn A <-> F Fn B ) )
2	1	anbi1d 465	. 2 |- ( A = B -> ( ( F Fn A /\ ran F C_ C ) <-> ( F Fn B /\ ran F C_ C ) ) )
3		df-f 5356	. 2 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
4		df-f 5356	. 2 |- ( F : B --> C <-> ( F Fn B /\ ran F C_ C ) )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( F : A --> C <-> F : B --> C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   C_ wss 3211   ran crn 4750   Fn wfn 5347   -->wf 5348

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-fn 5355  df-f 5356

This theorem is referenced by:  feq23  5494  feq2d  5496  feq2i  5502  f00  5559  f0dom0  5561  f1eq2  5569  fressnfv  5871  tfrcllemsucfn  6584  tfrcllemsucaccv  6585  tfrcllembxssdm  6587  tfrcllembfn  6588  tfrcllemaccex  6592  tfrcllemres  6593  tfrcldm  6594  tfrcl  6595  mapvalg  6892  map0g  6922  ac6sfi  7155  isomni  7427  ismkv  7444  iswomni  7456  isghm  13960