Theorem restlly 22634

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 3921	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ Top)
3		simprl 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝑗)
4		vex 3436	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	4	pwid 4557	. . . . . . . 8 ⊢ 𝑥 ∈ 𝒫 𝑥
6	5	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝒫 𝑥)
7	3, 6	elind 4128	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ (𝑗 ∩ 𝒫 𝑥))
8		simprr 770	. . . . . 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 468	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝐴)
11	10	adantrr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
12		elequ2 2121	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑥))
13		oveq2 7283	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑗 ↾t 𝑢) = (𝑗 ↾t 𝑥))
14	13	eleq1d 2823	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → ((𝑗 ↾t 𝑢) ∈ 𝐴 ↔ (𝑗 ↾t 𝑥) ∈ 𝐴))
15	12, 14	anbi12d 631	. . . . . . 7 ⊢ (𝑢 = 𝑥 → ((𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) ↔ (𝑦 ∈ 𝑥 ∧ (𝑗 ↾t 𝑥) ∈ 𝐴)))
16	15	rspcev 3561	. . . . . 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 834	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
18	17	ralrimivva 3123	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
19		islly 22619	. . . 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 413	. 2 ⊢ (𝜑 → (𝑗 ∈ 𝐴 → 𝑗 ∈ Locally 𝐴))
22	21	ssrdv 3927	1 ⊢ (𝜑 → 𝐴 ⊆ Locally 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  (class class class)co 7275   ↾_t crest 17131  Topctop 22042  Locally clly 22615

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-lly 22617

This theorem is referenced by:  llyidm  22639  nllyidm  22640  toplly  22641  hauslly  22643  lly1stc  22647