Theorem llyi 22978

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 22972	. . . 4 ⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
2	1	simprbi 498	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
3		pweq 4617	. . . . . . 7 ⊢ (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
4	3	ineq2d 4213	. . . . . 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 3334	. . . 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 3612	. . 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 581	. 2 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽) → ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
9		eleq1 2822	. . . . . . 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 470	. . . . . 6 ⊢ (((𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑈 ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))
13		elin 3965	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ∈ 𝒫 𝑈))
14		velpw 4608	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝒫 𝑈 ↔ 𝑢 ⊆ 𝑈)
15	14	anbi2i 624	. . . . . . . 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 625	. . . . . 6 ⊢ ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ ((𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
18		3anass 1096	. . . . . . 7 ⊢ ((𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ (𝑢 ⊆ 𝑈 ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
19	18	anbi2i 624	. . . . . 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 3612	. 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 1779	1 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ∩ cin 3948   ⊆ wss 3949  𝒫 cpw 4603  (class class class)co 7409   ↾_t crest 17366  Topctop 22395  Locally clly 22968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-lly 22970

This theorem is referenced by:  llynlly  22981  islly2  22988  llyrest  22989  llyidm  22992  nllyidm  22993  lly1stc  23000  dislly  23001  txlly  23140  cvmlift2lem10  34303