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Theorem nffn 5947
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 5853 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2 nffn.1 . . . 4 𝑥𝐹
32nffun 5872 . . 3 𝑥Fun 𝐹
42nfdm 5331 . . . 4 𝑥dom 𝐹
5 nffn.2 . . . 4 𝑥𝐴
64, 5nfeq 2778 . . 3 𝑥dom 𝐹 = 𝐴
73, 6nfan 1830 . 2 𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
81, 7nfxfr 1777 1 𝑥 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wnf 1705  wnfc 2754  dom cdm 5079  Fun wfun 5844   Fn wfn 5845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-fun 5852  df-fn 5853
This theorem is referenced by:  nff  6000  nffo  6073  feqmptdf  6209  nfixp  7872  nfixp1  7873  bnj1463  30823  choicefi  38852  stoweidlem31  39542  stoweidlem35  39546  stoweidlem59  39570
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