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Theorem nfdfat 47139
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, , 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 47131 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 nfdfat.2 . . . 4 𝑥𝐴
3 nfdfat.1 . . . . 5 𝑥𝐹
43nfdm 5962 . . . 4 𝑥dom 𝐹
52, 4nfel 2920 . . 3 𝑥 𝐴 ∈ dom 𝐹
62nfsn 4707 . . . . 5 𝑥{𝐴}
73, 6nfres 5999 . . . 4 𝑥(𝐹 ↾ {𝐴})
87nffun 6589 . . 3 𝑥Fun (𝐹 ↾ {𝐴})
95, 8nfan 1899 . 2 𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
101, 9nfxfr 1853 1 𝑥 𝐹 defAt 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395  wnf 1783  wcel 2108  wnfc 2890  {csn 4626  dom cdm 5685  cres 5687  Fun wfun 6555   defAt wdfat 47128
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-in 3958  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-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-fun 6563  df-dfat 47131
This theorem is referenced by:  nfafv  47148  nfafv2  47230
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