Theorem feq2 5388

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 5344	. . 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 5259	. 2 |- ( F : A --> C <-> ( F Fn A /\ ran F C_ C ) )
4		df-f 5259	. 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 3154   ran crn 4661   Fn wfn 5250   -->wf 5251

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-fn 5258  df-f 5259

This theorem is referenced by:  feq23  5390  feq2d  5392  feq2i  5398  f00  5446  f0dom0  5448  f1eq2  5456  fressnfv  5746  tfrcllemsucfn  6408  tfrcllemsucaccv  6409  tfrcllembxssdm  6411  tfrcllembfn  6412  tfrcllemaccex  6416  tfrcllemres  6417  tfrcldm  6418  tfrcl  6419  mapvalg  6714  map0g  6744  ac6sfi  6956  isomni  7197  ismkv  7214  iswomni  7226  isghm  13316