Theorem feq1 5455

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 5408	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4950	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3253	. . 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 5321	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5321	. 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 1395   C_ wss 3197   ran crn 4719   Fn wfn 5312   -->wf 5313

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-f 5321

This theorem is referenced by:  feq1d  5459  feq1i  5465  f00  5516  f0bi  5517  f0dom0  5518  fconstg  5521  f1eq1  5525  fconst2g  5853  tfrcllemsucfn  6497  tfrcllemsucaccv  6498  tfrcllembxssdm  6500  tfrcllembfn  6501  tfrcllemex  6504  tfrcllemaccex  6505  tfrcllemres  6506  tfrcl  6508  elmapg  6806  ac6sfi  7056  updjud  7245  finomni  7303  exmidomni  7305  mkvprop  7321  1fv  10331  seqf1oglem2  10737  seqf1og  10738  iswrd  11068  isgrpinv  13582  isghm  13775  upxp  14940  txcn  14943  plyf  15405  dceqnconst  16387  dcapnconst  16388