Theorem feq1 5330

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 5286	. . 3 |- ( F = G -> ( F Fn A <-> G Fn A ) )
2		rneq 4838	. . . 4 |- ( F = G -> ran F = ran G )
3	2	sseq1d 3176	. . 3 |- ( F = G -> ( ran F C_ B <-> ran G C_ B ) )
4	1, 3	anbi12d 470	. 2 |- ( F = G -> ( ( F Fn A /\ ran F C_ B ) <-> ( G Fn A /\ ran G C_ B ) ) )
5		df-f 5202	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
6		df-f 5202	. 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 1348   C_ wss 3121   ran crn 4612   Fn wfn 5193   -->wf 5194

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202

This theorem is referenced by:  feq1d  5334  feq1i  5340  f00  5389  f0bi  5390  f0dom0  5391  fconstg  5394  f1eq1  5398  fconst2g  5711  tfrcllemsucfn  6332  tfrcllemsucaccv  6333  tfrcllembxssdm  6335  tfrcllembfn  6336  tfrcllemex  6339  tfrcllemaccex  6340  tfrcllemres  6341  tfrcl  6343  elmapg  6639  ac6sfi  6876  updjud  7059  finomni  7116  exmidomni  7118  mkvprop  7134  1fv  10095  isgrpinv  12756  upxp  13066  txcn  13069  dceqnconst  14091  dcapnconst  14092