Theorem feq1 5386

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 5342	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4889	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3208	. . 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 5258	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5258	. 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 3153   ran crn 4660   Fn wfn 5249   -->wf 5250

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 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-fun 5256  df-fn 5257  df-f 5258

This theorem is referenced by:  feq1d  5390  feq1i  5396  f00  5445  f0bi  5446  f0dom0  5447  fconstg  5450  f1eq1  5454  fconst2g  5773  tfrcllemsucfn  6406  tfrcllemsucaccv  6407  tfrcllembxssdm  6409  tfrcllembfn  6410  tfrcllemex  6413  tfrcllemaccex  6414  tfrcllemres  6415  tfrcl  6417  elmapg  6715  ac6sfi  6954  updjud  7141  finomni  7199  exmidomni  7201  mkvprop  7217  1fv  10205  seqf1oglem2  10591  seqf1og  10592  iswrd  10916  isgrpinv  13126  isghm  13313  upxp  14440  txcn  14443  plyf  14883  dceqnconst  15550  dcapnconst  15551