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Theorem nfdfat 43333
 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 43325 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 nfdfat.2 . . . 4 𝑥𝐴
3 nfdfat.1 . . . . 5 𝑥𝐹
43nfdm 5826 . . . 4 𝑥dom 𝐹
52, 4nfel 2995 . . 3 𝑥 𝐴 ∈ dom 𝐹
62nfsn 4646 . . . . 5 𝑥{𝐴}
73, 6nfres 5858 . . . 4 𝑥(𝐹 ↾ {𝐴})
87nffun 6381 . . 3 𝑥Fun (𝐹 ↾ {𝐴})
95, 8nfan 1899 . 2 𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
101, 9nfxfr 1852 1 𝑥 𝐹 defAt 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 398  Ⅎwnf 1783   ∈ wcel 2113  Ⅎwnfc 2964  {csn 4570  dom cdm 5558   ↾ cres 5560  Fun wfun 6352   defAt wdfat 43322 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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-res 5570  df-fun 6360  df-dfat 43325 This theorem is referenced by:  nfafv  43342  nfafv2  43424
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