Theorem nfdfat 47721

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 47713	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		nfdfat.2	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfdfat.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
4	3	nfdm 5927	. . . 4 ⊢ Ⅎ𝑥dom 𝐹
5	2, 4	nfel 2938	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝐹
6	2	nfsn 4666	. . . . 5 ⊢ Ⅎ𝑥{𝐴}
7	3, 6	nfres 5967	. . . 4 ⊢ Ⅎ𝑥(𝐹 ↾ {𝐴})
8	7	nffun 6544	. . 3 ⊢ Ⅎ𝑥Fun (𝐹 ↾ {𝐴})
9	5, 8	nfan 1919	. 2 ⊢ Ⅎ𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	1, 9	nfxfr 1873	1 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴

Syntax hints:   ∧ wa 399  Ⅎwnf 1803   ∈ wcel 2142  Ⅎwnfc 2909  {csn 4582  dom cdm 5647   ↾ cres 5649  Fun wfun 6515   defAt wdfat 47710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-res 5659  df-fun 6523  df-dfat 47713

This theorem is referenced by:  nfafv  47730  nfafv2  47812