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Theorem nffn 6667
Description: Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
Hypotheses
Ref Expression
nffn.1 𝑥𝐹
nffn.2 𝑥𝐴
Assertion
Ref Expression
nffn 𝑥 𝐹 Fn 𝐴

Proof of Theorem nffn
StepHypRef Expression
1 df-fn 6564 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2 nffn.1 . . . 4 𝑥𝐹
32nffun 6589 . . 3 𝑥Fun 𝐹
42nfdm 5962 . . . 4 𝑥dom 𝐹
5 nffn.2 . . . 4 𝑥𝐴
64, 5nfeq 2919 . . 3 𝑥dom 𝐹 = 𝐴
73, 6nfan 1899 . 2 𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
81, 7nfxfr 1853 1 𝑥 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wnf 1783  wnfc 2890  dom cdm 5685  Fun wfun 6555   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-fun 6563  df-fn 6564
This theorem is referenced by:  nff  6732  nffo  6819  feqmptdf  6979  nfixpw  8956  nfixp  8957  nfixp1  8958  bnj1463  35069  choicefi  45205  stoweidlem31  46046  stoweidlem35  46050  stoweidlem59  46074
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