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Theorem nffn 5506
Description: Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
Hypotheses
Ref Expression
nffn.1  |-  F/_ x F
nffn.2  |-  F/_ x A
Assertion
Ref Expression
nffn  |-  F/ x  F  Fn  A

Proof of Theorem nffn
StepHypRef Expression
1 df-fn 5420 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 nffn.1 . . . 4  |-  F/_ x F
32nffun 5439 . . 3  |-  F/ x Fun  F
42nfdm 5080 . . . 4  |-  F/_ x dom  F
5 nffn.2 . . . 4  |-  F/_ x A
64, 5nfeq 2585 . . 3  |-  F/ x dom  F  =  A
73, 6nfan 1861 . 2  |-  F/ x
( Fun  F  /\  dom  F  =  A )
81, 7nfxfr 1615 1  |-  F/ x  F  Fn  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369   F/wnf 1589   F/_wnfc 2565   dom cdm 4839   Fun wfun 5411    Fn wfn 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2719  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-fun 5419  df-fn 5420
This theorem is referenced by:  nff  5554  nffo  5618  nfixp  7281  nfixp1  7282  feqmptdf  25977  stoweidlem31  29824  stoweidlem35  29828  stoweidlem59  29852  bnj1463  32044
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