Theorem feq1 5465

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 5418	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4959	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3256	. . 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 5330	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5330	. 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 1397   C_ wss 3200   ran crn 4726   Fn wfn 5321   -->wf 5322

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330

This theorem is referenced by:  feq1d  5469  feq1i  5475  f00  5528  f0bi  5529  f0dom0  5530  fconstg  5533  f1eq1  5537  fconst2g  5869  tfrcllemsucfn  6519  tfrcllemsucaccv  6520  tfrcllembxssdm  6522  tfrcllembfn  6523  tfrcllemex  6526  tfrcllemaccex  6527  tfrcllemres  6528  tfrcl  6530  elmapg  6830  ac6sfi  7087  updjud  7281  finomni  7339  exmidomni  7341  mkvprop  7357  1fv  10374  seqf1oglem2  10783  seqf1og  10784  iswrd  11119  isgrpinv  13655  isghm  13848  upxp  15015  txcn  15018  plyf  15480  griedg0prc  16120  dceqnconst  16716  dcapnconst  16717