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Theorem nffn 6597
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 6495 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2 nffn.1 . . . 4 𝑥𝐹
32nffun 6520 . . 3 𝑥Fun 𝐹
42nfdm 5903 . . . 4 𝑥dom 𝐹
5 nffn.2 . . . 4 𝑥𝐴
64, 5nfeq 2919 . . 3 𝑥dom 𝐹 = 𝐴
73, 6nfan 1903 . 2 𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
81, 7nfxfr 1856 1 𝑥 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wnf 1786  wnfc 2886  dom cdm 5631  Fun wfun 6486   Fn wfn 6487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-fun 6494  df-fn 6495
This theorem is referenced by:  nff  6660  nffo  6751  feqmptdf  6908  nfixpw  8788  nfixp  8789  nfixp1  8790  bnj1463  33447  choicefi  43216  stoweidlem31  44063  stoweidlem35  44067  stoweidlem59  44091
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