Theorem feq1 5456

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 5409	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4951	. . . 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 5322	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5322	. 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 4720   Fn wfn 5313   -->wf 5314

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 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-fun 5320  df-fn 5321  df-f 5322

This theorem is referenced by:  feq1d  5460  feq1i  5466  f00  5519  f0bi  5520  f0dom0  5521  fconstg  5524  f1eq1  5528  fconst2g  5858  tfrcllemsucfn  6505  tfrcllemsucaccv  6506  tfrcllembxssdm  6508  tfrcllembfn  6509  tfrcllemex  6512  tfrcllemaccex  6513  tfrcllemres  6514  tfrcl  6516  elmapg  6816  ac6sfi  7068  updjud  7260  finomni  7318  exmidomni  7320  mkvprop  7336  1fv  10347  seqf1oglem2  10754  seqf1og  10755  iswrd  11086  isgrpinv  13602  isghm  13795  upxp  14961  txcn  14964  plyf  15426  dceqnconst  16488  dcapnconst  16489