Theorem fneq1 5409

Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
fneq1	|- ( F = G -> ( F Fn A <-> G Fn A ) )

Proof of Theorem fneq1

Step	Hyp	Ref	Expression
1		funeq 5338	. . 3 |- ( F = G -> ( Fun F <-> Fun G ) )
2		dmeq 4923	. . . 4 |- ( F = G -> dom F = dom G )
3	2	eqeq1d 2238	. . 3 |- ( F = G -> ( dom F = A <-> dom G = A ) )
4	1, 3	anbi12d 473	. 2 |- ( F = G -> ( ( Fun F /\ dom F = A ) <-> ( Fun G /\ dom G = A ) ) )
5		df-fn 5321	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
6		df-fn 5321	. 2 |- ( G Fn A <-> ( Fun G /\ dom G = A ) )
7	4, 5, 6	3bitr4g 223	1 |- ( F = G -> ( F Fn A <-> G Fn A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   dom cdm 4719   Fun wfun 5312   Fn wfn 5313

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-fun 5320  df-fn 5321

This theorem is referenced by:  fneq1d  5411  fneq1i  5415  fn0  5443  feq1  5456  foeq1  5544  f1ocnv  5585  mpteqb  5725  eufnfv  5870  uchoice  6283  tfr0dm  6468  tfrlemiex  6477  tfr1onlemsucfn  6486  tfr1onlemsucaccv  6487  tfr1onlembxssdm  6489  tfr1onlembfn  6490  tfr1onlemex  6493  tfr1onlemaccex  6494  tfr1onlemres  6495  mapval2  6825  elixp2  6849  ixpfn  6851  elixpsn  6882  cc2lem  7452  cc3  7454  lmodfopnelem1  14288