Theorem llyi 21277

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 21271	. . . 4 ⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
2	1	simprbi 480	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
3		pweq 4161	. . . . . . 7 ⊢ (𝑥 = 𝑈 → 𝒫 𝑥 = 𝒫 𝑈)
4	3	ineq2d 3814	. . . . . 6 ⊢ (𝑥 = 𝑈 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑈))
5	4	rexeqdv 3145	. . . . 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 3146	. . . 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 3308	. . 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 488	. 2 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽) → ∀𝑦 ∈ 𝑈 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑈)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
9		eleq1 2689	. . . . . . 7 ⊢ (𝑦 = 𝑃 → (𝑦 ∈ 𝑢 ↔ 𝑃 ∈ 𝑢))
10	9	anbi1d 741	. . . . . 6 ⊢ (𝑦 = 𝑃 → ((𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
11	10	anbi2d 740	. . . . 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 681	. . . . . 6 ⊢ (((𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑈 ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))))
13		elin 3796	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ∈ 𝒫 𝑈))
14		selpw 4165	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝒫 𝑈 ↔ 𝑢 ⊆ 𝑈)
15	14	anbi2i 730	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝐽 ∧ 𝑢 ∈ 𝒫 𝑈) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈))
16	13, 15	bitri 264	. . . . . . 7 ⊢ (𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ↔ (𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈))
17	16	anbi1i 731	. . . . . 6 ⊢ ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ ((𝑢 ∈ 𝐽 ∧ 𝑢 ⊆ 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
18		3anass 1042	. . . . . . 7 ⊢ ((𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴) ↔ (𝑢 ⊆ 𝑈 ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
19	18	anbi2i 730	. . . . . 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 292	. . . . 5 ⊢ ((𝑢 ∈ (𝐽 ∩ 𝒫 𝑈) ∧ (𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)) ↔ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴)))
21	11, 20	syl6bb 276	. . . 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 3048	. . 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 3308	. 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 1701	1 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑈 ∧ 𝑃 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  (class class class)co 6650   ↾_t crest 16081  Topctop 20698  Locally clly 21267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-lly 21269

This theorem is referenced by:  llynlly  21280  islly2  21287  llyrest  21288  llyidm  21291  nllyidm  21292  lly1stc  21299  dislly  21300  txlly  21439  cvmlift2lem10  31294