Theorem feq1 5491

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 5444	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4984	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3267	. . 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 5356	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5356	. 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 1398   C_ wss 3211   ran crn 4750   Fn wfn 5347   -->wf 5348

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-fun 5354  df-fn 5355  df-f 5356

This theorem is referenced by:  feq1d  5495  feq1i  5501  f00  5559  f0bi  5560  f0dom0  5561  fconstg  5564  f1eq1  5568  fconst2g  5899  tfrcllemsucfn  6584  tfrcllemsucaccv  6585  tfrcllembxssdm  6587  tfrcllembfn  6588  tfrcllemex  6591  tfrcllemaccex  6592  tfrcllemres  6593  tfrcl  6595  elmapg  6895  ac6sfi  7155  updjud  7373  finomni  7431  exmidomni  7433  mkvprop  7449  1fv  10473  seqf1oglem2  10882  seqf1og  10883  iswrd  11226  isgrpinv  13767  isghm  13960  upxp  15137  txcn  15140  plyf  15602  griedg0prc  16245  dceqnconst  16846  dcapnconst  16847