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Theorem nffn 5690
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 5604 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
2 nffn.1 . . . 4  |-  F/_ x F
32nffun 5623 . . 3  |-  F/ x Fun  F
42nfdm 5096 . . . 4  |-  F/_ x dom  F
5 nffn.2 . . . 4  |-  F/_ x A
64, 5nfeq 2602 . . 3  |-  F/ x dom  F  =  A
73, 6nfan 1986 . 2  |-  F/ x
( Fun  F  /\  dom  F  =  A )
81, 7nfxfr 1692 1  |-  F/ x  F  Fn  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   F/wnf 1663   F/_wnfc 2577   dom cdm 4854   Fun wfun 5595    Fn wfn 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-fun 5603  df-fn 5604
This theorem is referenced by:  nff  5742  nffo  5809  nfixp  7549  nfixp1  7550  feqmptdf  28106  bnj1463  29660  stoweidlem31  37476  stoweidlem35  37480  stoweidlem59  37504
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