Theorem nllyi 23532

Description: The property of an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
nllyi	⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))

Distinct variable groups:   𝑢,𝐴   𝑢,𝑃   𝑢,𝑈   𝑢,𝐽

Proof of Theorem nllyi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnlly 23526	. . . 4 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
2	1	simprbi 501	. . 3 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)
3		pweq 4569	. . . . . . 7 ⊢ (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
4	3	ineq2d 4172	. . . . . 6 ⊢ (𝑥 = 𝑈 → (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈))
5	4	rexeqdv 3321	. . . . 5 ⊢ (𝑥 = 𝑈 → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽 ↾t 𝑢) ∈ 𝐴))
6	5	raleqbi1dv 3330	. . . 4 ⊢ (𝑥 = 𝑈 → (∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴 ↔ ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽 ↾t 𝑢) ∈ 𝐴))
7	6	rspccva 3580	. . 3 ⊢ ((∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴 ∧ 𝑈 ∈ 𝐽) → ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽 ↾t 𝑢) ∈ 𝐴)
8	2, 7	sylan 589	. 2 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽) → ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽 ↾t 𝑢) ∈ 𝐴)
9		elin 3920	. . . . . . 7 ⊢ (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ∧ 𝑢 ∈ 𝒫 𝑈))
10		sneq 4592	. . . . . . . . . 10 ⊢ (𝑦 = 𝑃 → {𝑦} = {𝑃})
11	10	fveq2d 6871	. . . . . . . . 9 ⊢ (𝑦 = 𝑃 → ((nei‘𝐽)‘{𝑦}) = ((nei‘𝐽)‘{𝑃}))
12	11	eleq2d 2848	. . . . . . . 8 ⊢ (𝑦 = 𝑃 → (𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ↔ 𝑢 ∈ ((nei‘𝐽)‘{𝑃})))
13		velpw 4560	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝒫 𝑈 ↔ 𝑢 ⊆ 𝑈)
14	13	a1i 11	. . . . . . . 8 ⊢ (𝑦 = 𝑃 → (𝑢 ∈ 𝒫 𝑈 ↔ 𝑢 ⊆ 𝑈))
15	12, 14	anbi12d 641	. . . . . . 7 ⊢ (𝑦 = 𝑃 → ((𝑢 ∈ ((nei‘𝐽)‘{𝑦}) ∧ 𝑢 ∈ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢 ⊆ 𝑈)))
16	9, 15	bitrid 285	. . . . . 6 ⊢ (𝑦 = 𝑃 → (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢 ⊆ 𝑈)))
17	16	anbi1d 640	. . . . 5 ⊢ (𝑦 = 𝑃 → ((𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ ((𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢 ⊆ 𝑈) ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
18		anass 472	. . . . 5 ⊢ (((𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ 𝑢 ⊆ 𝑈) ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑢 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
19	17, 18	bitrdi 289	. . . 4 ⊢ (𝑦 = 𝑃 → ((𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈) ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ (𝑢 ∈ ((nei‘𝐽)‘{𝑃}) ∧ (𝑢 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))
20	19	rexbidv2 3182	. . 3 ⊢ (𝑦 = 𝑃 → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽 ↾t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
21	20	rspccva 3580	. 2 ⊢ ((∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑈)(𝐽 ↾t 𝑢) ∈ 𝐴 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
22	8, 21	stoic3 1796	1 ⊢ ((𝐽 ∈ 𝑛-Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑃})(𝑢 ⊆ 𝑈 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4555  {csn 4582  ‘cfv 6521  (class class class)co 7396   ↾_t crest 17449  Topctop 22950  neicnei 23154  𝑛-Locally cnlly 23522

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-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-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  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-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-ov 7399  df-nlly 23524

This theorem is referenced by:  nlly2i  23533  llycmpkgen  23609