Theorem feq1 5263

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 5219	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4774	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3131	. . 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 5135	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5135	. 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 1332   C_ wss 3076   ran crn 4548   Fn wfn 5126   -->wf 5127

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-fun 5133  df-fn 5134  df-f 5135

This theorem is referenced by:  feq1d  5267  feq1i  5273  f00  5322  f0bi  5323  f0dom0  5324  fconstg  5327  f1eq1  5331  fconst2g  5643  tfrcllemsucfn  6258  tfrcllemsucaccv  6259  tfrcllembxssdm  6261  tfrcllembfn  6262  tfrcllemex  6265  tfrcllemaccex  6266  tfrcllemres  6267  tfrcl  6269  elmapg  6563  ac6sfi  6800  updjud  6975  finomni  7020  exmidomni  7022  mkvprop  7040  1fv  9947  upxp  12480  txcn  12483  dceqnconst  13423