Theorem feq1 5393

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 5347	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4894	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3213	. . 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 5263	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5263	. 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 1364   C_ wss 3157   ran crn 4665   Fn wfn 5254   -->wf 5255

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-fun 5261  df-fn 5262  df-f 5263

This theorem is referenced by:  feq1d  5397  feq1i  5403  f00  5452  f0bi  5453  f0dom0  5454  fconstg  5457  f1eq1  5461  fconst2g  5780  tfrcllemsucfn  6420  tfrcllemsucaccv  6421  tfrcllembxssdm  6423  tfrcllembfn  6424  tfrcllemex  6427  tfrcllemaccex  6428  tfrcllemres  6429  tfrcl  6431  elmapg  6729  ac6sfi  6968  updjud  7157  finomni  7215  exmidomni  7217  mkvprop  7233  1fv  10231  seqf1oglem2  10629  seqf1og  10630  iswrd  10954  isgrpinv  13256  isghm  13449  upxp  14592  txcn  14595  plyf  15057  dceqnconst  15791  dcapnconst  15792