Theorem feq1 5407

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 5361	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4904	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3221	. . 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 5274	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5274	. 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 1372   C_ wss 3165   ran crn 4675   Fn wfn 5265   -->wf 5266

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-fun 5272  df-fn 5273  df-f 5274

This theorem is referenced by:  feq1d  5411  feq1i  5417  f00  5466  f0bi  5467  f0dom0  5468  fconstg  5471  f1eq1  5475  fconst2g  5798  tfrcllemsucfn  6438  tfrcllemsucaccv  6439  tfrcllembxssdm  6441  tfrcllembfn  6442  tfrcllemex  6445  tfrcllemaccex  6446  tfrcllemres  6447  tfrcl  6449  elmapg  6747  ac6sfi  6994  updjud  7183  finomni  7241  exmidomni  7243  mkvprop  7259  1fv  10260  seqf1oglem2  10663  seqf1og  10664  iswrd  10994  isgrpinv  13357  isghm  13550  upxp  14715  txcn  14718  plyf  15180  dceqnconst  15961  dcapnconst  15962