Theorem feq1 5511

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 5464	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 5004	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3277	. . 3 |- ( F = G -> ( ran F C_ B <-> ran G C_ B ) )
4	1, 3	anbi12d 477	. 2 |- ( F = G -> ( ( F Fn A /\ ran F C_ B ) <-> ( G Fn A /\ ran G C_ B ) ) )
5		df-f 5376	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5376	. 2 |- ( G : A --> B <-> ( G Fn A /\ ran G C_ B ) )
7	4, 5, 6	3bitr4g 223	1 |- ( F = G -> ( F : A --> B <-> G : A --> B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   C_ wss 3220   ran crn 4770   Fn wfn 5367   -->wf 5368

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376

This theorem is referenced by:  feq1d  5515  feq1i  5521  f00  5579  f0bi  5580  f0dom0  5581  fconstg  5584  f1eq1  5588  fconst2g  5921  tfrcllemsucfn  6614  tfrcllemsucaccv  6615  tfrcllembxssdm  6617  tfrcllembfn  6618  tfrcllemex  6621  tfrcllemaccex  6622  tfrcllemres  6623  tfrcl  6625  elmapg  6925  mapfset  6935  fsetsspwxp  6938  ac6sfi  7192  f1setfi  7307  updjud  7412  finomni  7470  exmidomni  7472  mkvprop  7488  1fv  10524  seqf1oglem2  10935  seqf1og  10936  iswrd  11284  isgrpinv  13836  isghm  14023  upxp  15296  txcn  15299  plyf  15761  griedg0prc  16405  dceqnconst  17015  dcapnconst  17016