Theorem restlly 23835

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 3913	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ Top)
3		simprl 783	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝑗)
4		vex 3458	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5	4	pwid 4584	. . . . . . . 8 ⊢ 𝑥 ∈ 𝒫 𝑥
6	5	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝒫 𝑥)
7	3, 6	elind 4100	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ (𝑗 ∩ 𝒫 𝑥))
8		simprr 785	. . . . . 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 473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 𝐴)
11	10	adantrr 730	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → (𝑗 ↾t 𝑥) ∈ 𝐴)
12		elequ2 2162	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑥))
13		oveq2 7454	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑗 ↾t 𝑢) = (𝑗 ↾t 𝑥))
14	13	eleq1d 2849	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → ((𝑗 ↾t 𝑢) ∈ 𝐴 ↔ (𝑗 ↾t 𝑥) ∈ 𝐴))
15	12, 14	anbi12d 644	. . . . . . 7 ⊢ (𝑢 = 𝑥 → ((𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴) ↔ (𝑦 ∈ 𝑥 ∧ (𝑗 ↾t 𝑥) ∈ 𝐴)))
16	15	rspcev 3580	. . . . . 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 850	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ (𝑥 ∈ 𝑗 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
18	17	ralrimivva 3209	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴))
19		islly 23820	. . . 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 595	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ Locally 𝐴)
21	20	ex 418	. 2 ⊢ (𝜑 → (𝑗 ∈ 𝐴 → 𝑗 ∈ Locally 𝐴))
22	21	ssrdv 3919	1 ⊢ (𝜑 → 𝐴 ⊆ Locally 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  (class class class)co 7446   ↾_t crest 17623  Topctop 23244  Locally clly 23816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-ov 7449  df-lly 23818

This theorem is used by:  llyidm  23840  nllyidm  23841  toplly  23842  hauslly  23844  lly1stc  23848