Theorem feq1 5320

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 5276	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4831	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3171	. . 3 |- ( F = G -> ( ran F C_ B <-> ran G C_ B ) )
4	1, 3	anbi12d 465	. 2 |- ( F = G -> ( ( F Fn A /\ ran F C_ B ) <-> ( G Fn A /\ ran G C_ B ) ) )
5		df-f 5192	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5192	. 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 1343   C_ wss 3116   ran crn 4605   Fn wfn 5183   -->wf 5184

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  feq1d  5324  feq1i  5330  f00  5379  f0bi  5380  f0dom0  5381  fconstg  5384  f1eq1  5388  fconst2g  5700  tfrcllemsucfn  6321  tfrcllemsucaccv  6322  tfrcllembxssdm  6324  tfrcllembfn  6325  tfrcllemex  6328  tfrcllemaccex  6329  tfrcllemres  6330  tfrcl  6332  elmapg  6627  ac6sfi  6864  updjud  7047  finomni  7104  exmidomni  7106  mkvprop  7122  1fv  10074  upxp  12912  txcn  12915  dceqnconst  13938  dcapnconst  13939