Theorem feq1 5299

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

Assertion

Ref	Expression
feq1	|- ( F = G -> ( F : A --> B <-> G : A --> B ) )

Proof of Theorem feq1

Step	Hyp	Ref	Expression
1		fneq1 5255	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4810	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3157	. . 3 |- ( F = G -> ( ran F C_ B <-> ran G C_ B ) )
4	1, 3	anbi12d 465	. 2 |- ( F = G -> ( ( F Fn A /\ ran F C_ B ) <-> ( G Fn A /\ ran G C_ B ) ) )
5		df-f 5171	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5171	. 2 |- ( G : A --> B <-> ( G Fn A /\ ran G C_ B ) )
7	4, 5, 6	3bitr4g 222	1 |- ( F = G -> ( F : A --> B <-> G : A --> B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   C_ wss 3102   ran crn 4584   Fn wfn 5162   -->wf 5163

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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-fun 5169  df-fn 5170  df-f 5171

This theorem is referenced by:  feq1d  5303  feq1i  5309  f00  5358  f0bi  5359  f0dom0  5360  fconstg  5363  f1eq1  5367  fconst2g  5679  tfrcllemsucfn  6294  tfrcllemsucaccv  6295  tfrcllembxssdm  6297  tfrcllembfn  6298  tfrcllemex  6301  tfrcllemaccex  6302  tfrcllemres  6303  tfrcl  6305  elmapg  6599  ac6sfi  6836  updjud  7016  finomni  7066  exmidomni  7068  mkvprop  7084  1fv  10020  upxp  12632  txcn  12635  dceqnconst  13592  dcapnconst  13593