Theorem llyrest 23493

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 23480	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2		resttop 23168	. . 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 23185	. . . . 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 1198	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝐽 ∈ Locally 𝐴)
7		simp2l 1200	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝐽)
8		simp3 1139	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9		llyi 23482	. . . . . . . . 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 1222	. . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . 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 3994	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 1226	. . . . . . . . . . . . . 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 23185	. . . . . . . . . . . . . 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 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 4605	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ 𝒫 𝑥 ↔ 𝑣 ⊆ 𝑥)
22	12, 21	sylibr 234	. . . . . . . . . . . 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 4200	. . . . . . . . . . 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 1223	. . . . . . . . . . 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 23173	. . . . . . . . . . . . 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 1224	. . . . . . . . . . . 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 2841	. . . . . . . . . . 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 515	. . . . . . . . . 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 412	. . . . . . . . 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 3164	. . . . . . . 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 1119	. . . . . 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 3146	. . . . 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 412	. . . 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 240	. . 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 3145	. 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 23476	. 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  (class class class)co 7431   ↾_t crest 17465  Topctop 22899  Locally clly 23472

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-lly 23474

This theorem is referenced by:  loclly  23495  llyidm  23496