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Theorem nfdfat 41735
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 41721 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 nfdfat.2 . . . 4 𝑥𝐴
3 nfdfat.1 . . . . 5 𝑥𝐹
43nfdm 5523 . . . 4 𝑥dom 𝐹
52, 4nfel 2916 . . 3 𝑥 𝐴 ∈ dom 𝐹
62nfsn 4387 . . . . 5 𝑥{𝐴}
73, 6nfres 5554 . . . 4 𝑥(𝐹 ↾ {𝐴})
87nffun 6073 . . 3 𝑥Fun (𝐹 ↾ {𝐴})
95, 8nfan 1978 . 2 𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
101, 9nfxfr 1928 1 𝑥 𝐹 defAt 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383  wnf 1857  wcel 2140  wnfc 2890  {csn 4322  dom cdm 5267  cres 5269  Fun wfun 6044   defAt wdfat 41718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-res 5279  df-fun 6052  df-dfat 41721
This theorem is referenced by:  nfafv  41741
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