Theorem restlly 22090

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 3966	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ Top)
3		simprl 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝑗)
4		vex 3497	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	4	pwid 4562	. . . . . . . 8 ⊢ 𝑥 ∈ 𝒫 𝑥
6	5	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝒫 𝑥)
7	3, 6	elind 4170	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ (𝑗 ∩ 𝒫 𝑥))
8		simprr 771	. . . . . 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 470	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝐴)
11	10	adantrr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
12		elequ2 2125	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑥))
13		oveq2 7163	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑗 ↾t 𝑢) = (𝑗 ↾t 𝑥))
14	13	eleq1d 2897	. . . . . . . 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 834	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
18	17	ralrimivva 3191	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
19		islly 22075	. . . 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 585	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ Locally 𝐴)
21	20	ex 415	. 2 ⊢ (𝜑 → (𝑗 ∈ 𝐴 → 𝑗 ∈ Locally 𝐴))
22	21	ssrdv 3972	1 ⊢ (𝜑 → 𝐴 ⊆ Locally 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ∩ cin 3934   ⊆ wss 3935  𝒫 cpw 4538  (class class class)co 7155   ↾_t crest 16693  Topctop 21500  Locally clly 22071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362  df-ov 7158  df-lly 22073

This theorem is referenced by:  llyidm  22095  nllyidm  22096  toplly  22097  hauslly  22099  lly1stc  22103