Theorem feq3 5252

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

Assertion

Ref	Expression
feq3	|- ( A = B -> ( F : C --> A <-> F : C --> B ) )

Proof of Theorem feq3

Step	Hyp	Ref	Expression
1		sseq2 3116	. . 3 |- ( A = B -> ( ran F C_ A <-> ran F C_ B ) )
2	1	anbi2d 459	. 2 |- ( A = B -> ( ( F Fn C /\ ran F C_ A ) <-> ( F Fn C /\ ran F C_ B ) ) )
3		df-f 5122	. 2 |- ( F : C --> A <-> ( F Fn C /\ ran F C_ A ) )
4		df-f 5122	. 2 |- ( F : C --> B <-> ( F Fn C /\ ran F C_ B ) )
5	2, 3, 4	3bitr4g 222	1 |- ( A = B -> ( F : C --> A <-> F : C --> B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   C_ wss 3066   ran crn 4535   Fn wfn 5113   -->wf 5114

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079  df-f 5122

This theorem is referenced by:  feq23  5253  feq3d  5256  feq123d  5258  fun2  5291  fconstg  5314  f1eq3  5320  fsng  5586  fsn2  5587  fsnunf  5613  mapvalg  6545  mapsn  6577  lmff  12407  txcn  12433