Theorem nfdfat 47101

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 47093	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		nfdfat.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfdfat.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
4	3	nfdm 5904	. . . 4 ⊢ Ⅎ𝑥dom 𝐹
5	2, 4	nfel 2906	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝐹
6	2	nfsn 4667	. . . . 5 ⊢ Ⅎ𝑥{𝐴}
7	3, 6	nfres 5941	. . . 4 ⊢ Ⅎ𝑥(𝐹 ↾ {𝐴})
8	7	nffun 6523	. . 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 2876  {csn 4585  dom cdm 5631   ↾ cres 5633  Fun wfun 6493   defAt wdfat 47090

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 2701

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 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-fun 6501  df-dfat 47093

This theorem is referenced by:  nfafv  47110  nfafv2  47192