Theorem feq2 5457

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 5410	. . 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 5322	. 2 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
4		df-f 5322	. 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 1395   C_ wss 3197   ran crn 4720   Fn wfn 5313   -->wf 5314

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-fn 5321  df-f 5322

This theorem is referenced by:  feq23  5459  feq2d  5461  feq2i  5467  f00  5517  f0dom0  5519  f1eq2  5527  fressnfv  5826  tfrcllemsucfn  6499  tfrcllemsucaccv  6500  tfrcllembxssdm  6502  tfrcllembfn  6503  tfrcllemaccex  6507  tfrcllemres  6508  tfrcldm  6509  tfrcl  6510  mapvalg  6805  map0g  6835  ac6sfi  7060  isomni  7303  ismkv  7320  iswomni  7332  isghm  13780