Theorem feq1 5250

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 5206	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4761	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3121	. . 3 |- ( F = G -> ( ran F C_ B <-> ran G C_ B ) )
4	1, 3	anbi12d 464	. 2 |- ( F = G -> ( ( F Fn A /\ ran F C_ B ) <-> ( G Fn A /\ ran G C_ B ) ) )
5		df-f 5122	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5122	. 2 |- ( G : A --> B <-> ( G Fn A /\ ran G C_ B ) )
7	4, 5, 6	3bitr4g 222	1 |- ( F = G -> ( F : A --> B <-> G : A --> B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   C_ wss 3066   ran crn 4535   Fn wfn 5113   -->wf 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122

This theorem is referenced by:  feq1d  5254  feq1i  5260  f00  5309  f0bi  5310  f0dom0  5311  fconstg  5314  f1eq1  5318  fconst2g  5628  tfrcllemsucfn  6243  tfrcllemsucaccv  6244  tfrcllembxssdm  6246  tfrcllembfn  6247  tfrcllemex  6250  tfrcllemaccex  6251  tfrcllemres  6252  tfrcl  6254  elmapg  6548  ac6sfi  6785  updjud  6960  finomni  7005  exmidomni  7007  mkvprop  7025  1fv  9909  upxp  12430  txcn  12433