Theorem restlly 23491

Description: If the property 𝐴 passes to open subspaces, then a space which is 𝐴 is also locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)

Hypotheses

Ref	Expression
restlly.1	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
restlly.2	⊢ (𝜑 → 𝐴 ⊆ Top)

Assertion

Ref	Expression
restlly	⊢ (𝜑 → 𝐴 ⊆ Locally 𝐴)

Distinct variable groups:   𝑥,𝑗,𝐴   𝜑,𝑗,𝑥

Proof of Theorem restlly

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

Step	Hyp	Ref	Expression
1		restlly.2	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ Top)
2	1	sselda 3983	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ Top)
3		simprl 771	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝑗)
4		vex 3484	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	4	pwid 4622	. . . . . . . 8 ⊢ 𝑥 ∈ 𝒫 𝑥
6	5	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝒫 𝑥)
7	3, 6	elind 4200	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ (𝑗 ∩ 𝒫 𝑥))
8		simprr 773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥)
9		restlly.1	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
10	9	anassrs 467	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝐴)
11	10	adantrr 717	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
12		elequ2 2123	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑥))
13		oveq2 7439	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑗 ↾t 𝑢) = (𝑗 ↾t 𝑥))
14	13	eleq1d 2826	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → ((𝑗 ↾t 𝑢) ∈ 𝐴 ↔ (𝑗 ↾t 𝑥) ∈ 𝐴))
15	12, 14	anbi12d 632	. . . . . . 7 ⊢ (𝑢 = 𝑥 → ((𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) ↔ (𝑦 ∈ 𝑥 ∧ (𝑗 ↾t 𝑥) ∈ 𝐴)))
16	15	rspcev 3622	. . . . . 6 ⊢ ((𝑥 ∈ (𝑗 ∩ 𝒫 𝑥) ∧ (𝑦 ∈ 𝑥 ∧ (𝑗 ↾t 𝑥) ∈ 𝐴)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
17	7, 8, 11, 16	syl12anc 837	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
18	17	ralrimivva 3202	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
19		islly 23476	. . . 4 ⊢ (𝑗 ∈ Locally 𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)))
20	2, 18, 19	sylanbrc 583	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ Locally 𝐴)
21	20	ex 412	. 2 ⊢ (𝜑 → (𝑗 ∈ 𝐴 → 𝑗 ∈ Locally 𝐴))
22	21	ssrdv 3989	1 ⊢ (𝜑 → 𝐴 ⊆ Locally 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ 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:  llyidm  23496  nllyidm  23497  toplly  23498  hauslly  23500  lly1stc  23504