Theorem llyi 23682

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 23676	. . . 4 ⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
2	1	simprbi 503	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
3		pweq 4578	. . . . . . 7 ⊢ (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
4	3	ineq2d 4173	. . . . . 6 ⊢ (𝑥 = 𝑈 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑈))
5	4	rexeqdv 3326	. . . . 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 3335	. . . 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 3582	. . 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 592	. 2 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽) → ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
9		eleq1 2853	. . . . . . 7 ⊢ (𝑦 = 𝑃 → (𝑦 ∈ 𝑢 ↔ 𝑃 ∈ 𝑢))
10	9	anbi1d 643	. . . . . 6 ⊢ (𝑦 = 𝑃 → ((𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
11	10	anbi2d 642	. . . . 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 474	. . . . . 6 ⊢ (((𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑈 ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))
13		elin 3922	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ∈ 𝒫 𝑈))
14		velpw 4569	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝒫 𝑈 ↔ 𝑢 ⊆ 𝑈)
15	14	anbi2i 635	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐽 ∧ 𝑢 ∈ 𝒫 𝑈) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈))
16	13, 15	bitri 278	. . . . . . 7 ⊢ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈))
17	16	anbi1i 636	. . . . . 6 ⊢ ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ ((𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
18		3anass 1111	. . . . . . 7 ⊢ ((𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ (𝑢 ⊆ 𝑈 ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
19	18	anbi2i 635	. . . . . 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 306	. . . . 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 290	. . . 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 3187	. . 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 3582	. 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 1809	1 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4564  (class class class)co 7419   ↾_t crest 17495  Topctop 23100  Locally clly 23672

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-lly 23674

This theorem is used by:  llynlly  23685  islly2  23692  llyrest  23693  llyidm  23696  nllyidm  23697  lly1stc  23704  dislly  23705  txlly  23844  cvmlift2lem10  35841