Theorem llyrest 23372

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 23359	. . 3 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2		resttop 23047	. . 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 23064	. . . . 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 1138	. . . . . . . . 9 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9		llyi 23361	. . . . . . . . 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 770	. . . . . . . . . . . . 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 3957	. . . . . . . . . . . . 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 23064	. . . . . . . . . . . . . 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 4568	. . . . . . . . . . . . 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 4163	. . . . . . . . . . 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 23052	. . . . . . . . . . . . 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 2828	. . . . . . . . . . 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 3143	. . . . . . . 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 1118	. . . . . 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 3125	. . . . 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 3124	. 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 23355	. 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 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  (class class class)co 7387   ↾_t crest 17383  Topctop 22780  Locally clly 23351

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-en 8919  df-fin 8922  df-fi 9362  df-rest 17385  df-topgen 17406  df-top 22781  df-topon 22798  df-bases 22833  df-lly 23353

This theorem is referenced by:  loclly  23374  llyidm  23375