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Theorem nfdfat 39859
 Description: Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g., for Fun/Rel, dom, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
nfdfat.1 𝑥𝐹
nfdfat.2 𝑥𝐴
Assertion
Ref Expression
nfdfat 𝑥 𝐹 defAt 𝐴

Proof of Theorem nfdfat
StepHypRef Expression
1 df-dfat 39845 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 nfdfat.2 . . . 4 𝑥𝐴
3 nfdfat.1 . . . . 5 𝑥𝐹
43nfdm 5288 . . . 4 𝑥dom 𝐹
52, 4nfel 2763 . . 3 𝑥 𝐴 ∈ dom 𝐹
62nfsn 4189 . . . . 5 𝑥{𝐴}
73, 6nfres 5319 . . . 4 𝑥(𝐹 ↾ {𝐴})
87nffun 5826 . . 3 𝑥Fun (𝐹 ↾ {𝐴})
95, 8nfan 1816 . 2 𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
101, 9nfxfr 1771 1 𝑥 𝐹 defAt 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  {csn 4125  dom cdm 5038   ↾ cres 5040  Fun wfun 5798   defAt wdfat 39842 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-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-fun 5806  df-dfat 39845 This theorem is referenced by:  nfafv  39865
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