Theorem nfdfat 47132

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	⊢ Ⅎ𝑥𝐹
nfdfat.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfdfat	⊢ Ⅎ𝑥 𝐹 defAt 𝐴

Proof of Theorem nfdfat

Step	Hyp	Ref	Expression
1		df-dfat 47124	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		nfdfat.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfdfat.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
4	3	nfdm 5918	. . . 4 ⊢ Ⅎ𝑥dom 𝐹
5	2, 4	nfel 2907	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝐹
6	2	nfsn 4674	. . . . 5 ⊢ Ⅎ𝑥{𝐴}
7	3, 6	nfres 5955	. . . 4 ⊢ Ⅎ𝑥(𝐹 ↾ {𝐴})
8	7	nffun 6542	. . 3 ⊢ Ⅎ𝑥Fun (𝐹 ↾ {𝐴})
9	5, 8	nfan 1899	. 2 ⊢ Ⅎ𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	1, 9	nfxfr 1853	1 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴

Syntax hints:   ∧ wa 395  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2877  {csn 4592  dom cdm 5641   ↾ cres 5643  Fun wfun 6508   defAt wdfat 47121

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-fun 6516  df-dfat 47124

This theorem is referenced by:  nfafv  47141  nfafv2  47223