Theorem restlly 23377

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 3949	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ Top)
3		simprl 770	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝑗)
4		vex 3454	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	4	pwid 4588	. . . . . . . 8 ⊢ 𝑥 ∈ 𝒫 𝑥
6	5	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝒫 𝑥)
7	3, 6	elind 4166	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ (𝑗 ∩ 𝒫 𝑥))
8		simprr 772	. . . . . 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 2124	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑥))
13		oveq2 7398	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑗 ↾t 𝑢) = (𝑗 ↾t 𝑥))
14	13	eleq1d 2814	. . . . . . . 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 3591	. . . . . 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 836	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
18	17	ralrimivva 3181	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
19		islly 23362	. . . 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 3955	1 ⊢ (𝜑 → 𝐴 ⊆ Locally 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ∩ cin 3916   ⊆ wss 3917  𝒫 cpw 4566  (class class class)co 7390   ↾_t crest 17390  Topctop 22787  Locally clly 23358

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-lly 23360

This theorem is referenced by:  llyidm  23382  nllyidm  23383  toplly  23384  hauslly  23386  lly1stc  23390