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Theorem nffn 5672
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 5585 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 nffn.1 . . . 4  |-  F/_ x F
32nffun 5604 . . 3  |-  F/ x Fun  F
42nfdm 5076 . . . 4  |-  F/_ x dom  F
5 nffn.2 . . . 4  |-  F/_ x A
64, 5nfeq 2603 . . 3  |-  F/ x dom  F  =  A
73, 6nfan 2011 . 2  |-  F/ x
( Fun  F  /\  dom  F  =  A )
81, 7nfxfr 1696 1  |-  F/ x  F  Fn  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1444   F/wnf 1667   F/_wnfc 2579   dom cdm 4834   Fun wfun 5576    Fn wfn 5577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-fun 5584  df-fn 5585
This theorem is referenced by:  nff  5724  nffo  5792  nfixp  7541  nfixp1  7542  feqmptdf  28258  bnj1463  29864  choicefi  37481  stoweidlem31  37892  stoweidlem35  37896  stoweidlem59  37920
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