Theorem feq2 5392

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 5348	. . 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 5263	. 2 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
4		df-f 5263	. 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 1364   C_ wss 3157   ran crn 4665   Fn wfn 5254   -->wf 5255

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-fn 5262  df-f 5263

This theorem is referenced by:  feq23  5394  feq2d  5396  feq2i  5402  f00  5450  f0dom0  5452  f1eq2  5460  fressnfv  5750  tfrcllemsucfn  6413  tfrcllemsucaccv  6414  tfrcllembxssdm  6416  tfrcllembfn  6417  tfrcllemaccex  6421  tfrcllemres  6422  tfrcldm  6423  tfrcl  6424  mapvalg  6719  map0g  6749  ac6sfi  6961  isomni  7204  ismkv  7221  iswomni  7233  isghm  13399