Theorem llyi 23482

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

Assertion

Ref	Expression
llyi	⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))

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

Proof of Theorem llyi

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

Step	Hyp	Ref	Expression
1		islly 23476	. . . 4 ⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
2	1	simprbi 496	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
3		pweq 4614	. . . . . . 7 ⊢ (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
4	3	ineq2d 4220	. . . . . 6 ⊢ (𝑥 = 𝑈 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑈))
5	4	rexeqdv 3327	. . . . 5 ⊢ (𝑥 = 𝑈 → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
6	5	raleqbi1dv 3338	. . . 4 ⊢ (𝑥 = 𝑈 → (∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
7	6	rspccva 3621	. . 3 ⊢ ((∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ∧ 𝑈 ∈ 𝐽) → ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
8	2, 7	sylan 580	. 2 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽) → ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
9		eleq1 2829	. . . . . . 7 ⊢ (𝑦 = 𝑃 → (𝑦 ∈ 𝑢 ↔ 𝑃 ∈ 𝑢))
10	9	anbi1d 631	. . . . . 6 ⊢ (𝑦 = 𝑃 → ((𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
11	10	anbi2d 630	. . . . 5 ⊢ (𝑦 = 𝑃 → ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))
12		anass 468	. . . . . 6 ⊢ (((𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑈 ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))
13		elin 3967	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ∈ 𝒫 𝑈))
14		velpw 4605	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝒫 𝑈 ↔ 𝑢 ⊆ 𝑈)
15	14	anbi2i 623	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐽 ∧ 𝑢 ∈ 𝒫 𝑈) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈))
16	13, 15	bitri 275	. . . . . . 7 ⊢ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈))
17	16	anbi1i 624	. . . . . 6 ⊢ ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ ((𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
18		3anass 1095	. . . . . . 7 ⊢ ((𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ (𝑢 ⊆ 𝑈 ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
19	18	anbi2i 623	. . . . . 6 ⊢ ((𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑈 ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))
20	12, 17, 19	3bitr4i 303	. . . . 5 ⊢ ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
21	11, 20	bitrdi 287	. . . 4 ⊢ (𝑦 = 𝑃 → ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))
22	21	rexbidv2 3175	. . 3 ⊢ (𝑦 = 𝑃 → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
23	22	rspccva 3621	. 2 ⊢ ((∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
24	8, 23	stoic3 1776	1 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  (class class class)co 7431   ↾_t crest 17465  Topctop 22899  Locally clly 23472

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-lly 23474

This theorem is referenced by:  llynlly  23485  islly2  23492  llyrest  23493  llyidm  23496  nllyidm  23497  lly1stc  23504  dislly  23505  txlly  23644  cvmlift2lem10  35317