Theorem feq1 5428

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 5381	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4924	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3230	. . 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 5294	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5294	. 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 1373   C_ wss 3174   ran crn 4694   Fn wfn 5285   -->wf 5286

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-fun 5292  df-fn 5293  df-f 5294

This theorem is referenced by:  feq1d  5432  feq1i  5438  f00  5489  f0bi  5490  f0dom0  5491  fconstg  5494  f1eq1  5498  fconst2g  5822  tfrcllemsucfn  6462  tfrcllemsucaccv  6463  tfrcllembxssdm  6465  tfrcllembfn  6466  tfrcllemex  6469  tfrcllemaccex  6470  tfrcllemres  6471  tfrcl  6473  elmapg  6771  ac6sfi  7021  updjud  7210  finomni  7268  exmidomni  7270  mkvprop  7286  1fv  10296  seqf1oglem2  10702  seqf1og  10703  iswrd  11033  isgrpinv  13501  isghm  13694  upxp  14859  txcn  14862  plyf  15324  dceqnconst  16201  dcapnconst  16202