Theorem feq1 5387

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 5343	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4890	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3209	. . 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 5259	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5259	. 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 3154   ran crn 4661   Fn wfn 5250   -->wf 5251

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-fun 5257  df-fn 5258  df-f 5259

This theorem is referenced by:  feq1d  5391  feq1i  5397  f00  5446  f0bi  5447  f0dom0  5448  fconstg  5451  f1eq1  5455  fconst2g  5774  tfrcllemsucfn  6408  tfrcllemsucaccv  6409  tfrcllembxssdm  6411  tfrcllembfn  6412  tfrcllemex  6415  tfrcllemaccex  6416  tfrcllemres  6417  tfrcl  6419  elmapg  6717  ac6sfi  6956  updjud  7143  finomni  7201  exmidomni  7203  mkvprop  7219  1fv  10208  seqf1oglem2  10594  seqf1og  10595  iswrd  10919  isgrpinv  13129  isghm  13316  upxp  14451  txcn  14454  plyf  14916  dceqnconst  15620  dcapnconst  15621