Theorem nfdfat 46536

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 46528	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		nfdfat.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfdfat.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
4	3	nfdm 5957	. . . 4 ⊢ Ⅎ𝑥dom 𝐹
5	2, 4	nfel 2914	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝐹
6	2	nfsn 4716	. . . . 5 ⊢ Ⅎ𝑥{𝐴}
7	3, 6	nfres 5991	. . . 4 ⊢ Ⅎ𝑥(𝐹 ↾ {𝐴})
8	7	nffun 6581	. . 3 ⊢ Ⅎ𝑥Fun (𝐹 ↾ {𝐴})
9	5, 8	nfan 1894	. 2 ⊢ Ⅎ𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	1, 9	nfxfr 1847	1 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴

Syntax hints:   ∧ wa 394  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2879  {csn 4632  dom cdm 5682   ↾ cres 5684  Fun wfun 6547   defAt wdfat 46525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-fun 6555  df-dfat 46528

This theorem is referenced by:  nfafv  46545  nfafv2  46627