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Theorem nffn 5533
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 5449 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 nffn.1 . . . 4  |-  F/_ x F
32nffun 5468 . . 3  |-  F/ x Fun  F
42nfdm 5103 . . . 4  |-  F/_ x dom  F
5 nffn.2 . . . 4  |-  F/_ x A
64, 5nfeq 2578 . . 3  |-  F/ x dom  F  =  A
73, 6nfan 1846 . 2  |-  F/ x
( Fun  F  /\  dom  F  =  A )
81, 7nfxfr 1579 1  |-  F/ x  F  Fn  A
Colors of variables: wff set class
Syntax hints:    /\ wa 359   F/wnf 1553    = wceq 1652   F/_wnfc 2558   dom cdm 4870   Fun wfun 5440    Fn wfn 5441
This theorem is referenced by:  nff  5581  nffo  5644  nfixp  7073  nfixp1  7074  feqmptdf  24067  stoweidlem31  27747  stoweidlem35  27751  stoweidlem59  27775  bnj1463  29361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-fun 5448  df-fn 5449
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