Theorem llyrest 22636

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 22623	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2		resttop 22311	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
3	1, 2	sylan 580	. 2 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top)
4		restopn2 22328	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
5	1, 4	sylan 580	. . . 4 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)))
6		simp1l 1196	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝐽 ∈ Locally 𝐴)
7		simp2l 1198	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝐽)
8		simp3 1137	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9		llyi 22625	. . . . . . . . 9 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ 𝐽 (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))
10	6, 7, 8, 9	syl3anc 1370	. . . . . . . 8 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ 𝐽 (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))
11		simprl 768	. . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . 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 1226	. . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . 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 22328	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑣 ∈ (𝐽 ↾t 𝐵) ↔ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝐵)))
19	16, 17, 18	syl2anc 584	. . . . . . . . . . . . 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 710	. . . . . . . . . . . 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 4538	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ 𝒫 𝑥 ↔ 𝑣 ⊆ 𝑥)
22	12, 21	sylibr 233	. . . . . . . . . . . 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 4128	. . . . . . . . . . 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 1221	. . . . . . . . . . 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 22316	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑣 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐽) → ((𝐽 ↾t 𝐵) ↾t 𝑣) = (𝐽 ↾t 𝑣))
26	16, 14, 17, 25	syl3anc 1370	. . . . . . . . . . . 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 1222	. . . . . . . . . . . 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 2839	. . . . . . . . . . 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 516	. . . . . . . . . 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 413	. . . . . . . . 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 3199	. . . . . . . 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 1117	. . . . . 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 3103	. . . . 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 413	. . . 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 239	. . 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 3102	. 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 22619	. 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 583	1 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Locally 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ 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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-en 8734  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-lly 22617

This theorem is referenced by:  loclly  22638  llyidm  22639