Theorem feq1 5391

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 5347	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4894	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3213	. . 3 |- ( F = G -> ( ran F C_ B <-> ran G C_ B ) )
4	1, 3	anbi12d 473	. 2 |- ( F = G -> ( ( F Fn A /\ ran F C_ B ) <-> ( G Fn A /\ ran G C_ B ) ) )
5		df-f 5263	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5263	. 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 1364   C_ wss 3157   ran crn 4665   Fn wfn 5254   -->wf 5255

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-fun 5261  df-fn 5262  df-f 5263

This theorem is referenced by:  feq1d  5395  feq1i  5401  f00  5450  f0bi  5451  f0dom0  5452  fconstg  5455  f1eq1  5459  fconst2g  5778  tfrcllemsucfn  6412  tfrcllemsucaccv  6413  tfrcllembxssdm  6415  tfrcllembfn  6416  tfrcllemex  6419  tfrcllemaccex  6420  tfrcllemres  6421  tfrcl  6423  elmapg  6721  ac6sfi  6960  updjud  7149  finomni  7207  exmidomni  7209  mkvprop  7225  1fv  10216  seqf1oglem2  10614  seqf1og  10615  iswrd  10939  isgrpinv  13196  isghm  13383  upxp  14518  txcn  14521  plyf  14983  dceqnconst  15714  dcapnconst  15715