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Theorem nffn 5901
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 5807 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2 nffn.1 . . . 4 𝑥𝐹
32nffun 5826 . . 3 𝑥Fun 𝐹
42nfdm 5288 . . . 4 𝑥dom 𝐹
5 nffn.2 . . . 4 𝑥𝐴
64, 5nfeq 2762 . . 3 𝑥dom 𝐹 = 𝐴
73, 6nfan 1816 . 2 𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴)
81, 7nfxfr 1771 1 𝑥 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wnf 1699  wnfc 2738  dom cdm 5038  Fun wfun 5798   Fn wfn 5799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807
This theorem is referenced by:  nff  5954  nffo  6027  feqmptdf  6161  nfixp  7813  nfixp1  7814  bnj1463  30377  choicefi  38387  stoweidlem31  38924  stoweidlem35  38928  stoweidlem59  38952
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