Theorem fneq1 5169

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 5101	. . 3 |- ( F = G -> ( Fun F <-> Fun G ) )
2		dmeq 4699	. . . 4 |- ( F = G -> dom F = dom G )
3	2	eqeq1d 2123	. . 3 |- ( F = G -> ( dom F = A <-> dom G = A ) )
4	1, 3	anbi12d 462	. 2 |- ( F = G -> ( ( Fun F /\ dom F = A ) <-> ( Fun G /\ dom G = A ) ) )
5		df-fn 5084	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
6		df-fn 5084	. 2 |- ( G Fn A <-> ( Fun G /\ dom G = A ) )
7	4, 5, 6	3bitr4g 222	1 |- ( F = G -> ( F Fn A <-> G Fn A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1314   dom cdm 4499   Fun wfun 5075   Fn wfn 5076

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-fun 5083  df-fn 5084

This theorem is referenced by:  fneq1d  5171  fneq1i  5175  fn0  5200  feq1  5213  foeq1  5299  f1ocnv  5336  mpteqb  5465  eufnfv  5602  tfr0dm  6173  tfrlemiex  6182  tfr1onlemsucfn  6191  tfr1onlemsucaccv  6192  tfr1onlembxssdm  6194  tfr1onlembfn  6195  tfr1onlemex  6198  tfr1onlemaccex  6199  tfr1onlemres  6200  mapval2  6526  elixp2  6550  ixpfn  6552  elixpsn  6583