Theorem feq1 5516

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 5469	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 5009	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3277	. . 3 |- ( F = G -> ( ran F C_ B <-> ran G C_ B ) )
4	1, 3	anbi12d 477	. 2 |- ( F = G -> ( ( F Fn A /\ ran F C_ B ) <-> ( G Fn A /\ ran G C_ B ) ) )
5		df-f 5381	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5381	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   C_ wss 3220   ran crn 4775   Fn wfn 5372   -->wf 5373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-fun 5379  df-fn 5380  df-f 5381

This theorem is used by:  feq1d  5520  feq1i  5526  f00  5584  f0bi  5585  f0dom0  5586  fconstg  5589  f1eq1  5593  fconst2g  5930  tfrcllemsucfn  6624  tfrcllemsucaccv  6625  tfrcllembxssdm  6627  tfrcllembfn  6628  tfrcllemex  6631  tfrcllemaccex  6632  tfrcllemres  6633  tfrcl  6635  elmapg  6935  mapfset  6945  fsetsspwxp  6948  ac6sfi  7202  f1setfi  7317  updjud  7422  finomni  7480  exmidomni  7482  mkvprop  7498  1fv  10546  seqf1oglem2  10957  seqf1og  10958  iswrd  11306  isgrpinv  13859  isghm  14046  upxp  15373  txcn  15376  plyf  15838  griedg0prc  16491  dceqnconst  17110  dcapnconst  17111