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Theorem nfdfat 45835
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 45827 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 nfdfat.2 . . . 4 𝑥𝐴
3 nfdfat.1 . . . . 5 𝑥𝐹
43nfdm 5951 . . . 4 𝑥dom 𝐹
52, 4nfel 2918 . . 3 𝑥 𝐴 ∈ dom 𝐹
62nfsn 4712 . . . . 5 𝑥{𝐴}
73, 6nfres 5984 . . . 4 𝑥(𝐹 ↾ {𝐴})
87nffun 6572 . . 3 𝑥Fun (𝐹 ↾ {𝐴})
95, 8nfan 1903 . 2 𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
101, 9nfxfr 1856 1 𝑥 𝐹 defAt 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 397  wnf 1786  wcel 2107  wnfc 2884  {csn 4629  dom cdm 5677  cres 5679  Fun wfun 6538   defAt wdfat 45824
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 2704
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 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-fun 6546  df-dfat 45827
This theorem is referenced by:  nfafv  45844  nfafv2  45926
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