Theorem llyrest 22382

Description: An open subspace of a locally 𝐴 space is also locally 𝐴. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
llyrest	⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Locally 𝐴)

Proof of Theorem llyrest

Dummy variables 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		llytop 22369	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2		resttop 22057	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
3	1, 2	sylan 583	. 2 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
4		restopn2 22074	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
5	1, 4	sylan 583	. . . 4 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
6		simp1l 1199	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝐽 ∈ Locally 𝐴)
7		simp2l 1201	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝐽)
8		simp3 1140	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9		llyi 22371	. . . . . . . . 9 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ 𝐽 (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))
10	6, 7, 8, 9	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ 𝐽 (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))
11		simprl 771	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝐽)
12		simprr1 1223	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ 𝑥)
13		simpl2r 1229	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑥 ⊆ 𝐵)
14	12, 13	sstrd 3911	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ 𝐵)
15	6, 1	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝐽 ∈ Top)
16	15	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝐽 ∈ Top)
17		simpl1r 1227	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝐵 ∈ 𝐽)
18		restopn2 22074	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑣 ∈ (𝐽 ↾t 𝐵) ↔ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝐵)))
19	16, 17, 18	syl2anc 587	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (𝐽 ↾t 𝐵) ↔ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝐵)))
20	11, 14, 19	mpbir2and 713	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝐽 ↾t 𝐵))
21		velpw 4518	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ 𝒫 𝑥 ↔ 𝑣 ⊆ 𝑥)
22	12, 21	sylibr 237	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥)
23	20, 22	elind 4108	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥))
24		simprr2 1224	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑦 ∈ 𝑣)
25		restabs 22062	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐽) → ((𝐽 ↾t 𝐵) ↾t 𝑣) = (𝐽 ↾t 𝑣))
26	16, 14, 17, 25	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → ((𝐽 ↾t 𝐵) ↾t 𝑣) = (𝐽 ↾t 𝑣))
27		simprr3 1225	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → (𝐽 ↾t 𝑣) ∈ 𝐴)
28	26, 27	eqeltrd 2838	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)
29	23, 24, 28	jca32 519	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)))
30	29	ex 416	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → ((𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴))))
31	30	reximdv2 3190	. . . . . . . 8 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → (∃𝑣 ∈ 𝐽 (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴) → ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)))
32	10, 31	mpd 15	. . . . . . 7 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴))
33	32	3expa 1120	. . . . . 6 ⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)) ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴))
34	33	ralrimiva 3105	. . . . 5 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)) → ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴))
35	34	ex 416	. . . 4 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → ((𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) → ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)))
36	5, 35	sylbid 243	. . 3 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) → ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)))
37	36	ralrimiv 3104	. 2 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → ∀𝑥 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑥 ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴))
38		islly 22365	. 2 ⊢ ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ((𝐽 ↾t 𝐵) ∈ Top ∧ ∀𝑥 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑥 ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)))
39	3, 37, 38	sylanbrc 586	1 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Locally 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062   ∩ cin 3865   ⊆ wss 3866  𝒫 cpw 4513  (class class class)co 7213   ↾_t crest 16925  Topctop 21790  Locally clly 22361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-en 8627  df-fin 8630  df-fi 9027  df-rest 16927  df-topgen 16948  df-top 21791  df-topon 21808  df-bases 21843  df-lly 22363

This theorem is referenced by:  loclly  22384  llyidm  22385