Theorem subislly 22378

Description: The property of a subspace being locally 𝐴. (Contributed by Mario Carneiro, 10-Mar-2015.)

Assertion

Ref	Expression
subislly	⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))

Distinct variable groups:   𝑥,𝑢,𝑦,𝐴   𝑢,𝐵,𝑥,𝑦   𝑢,𝐽,𝑥,𝑦   𝑢,𝑉,𝑥,𝑦

Proof of Theorem subislly

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resttop 22057	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝐽 ↾t 𝐵) ∈ Top)
2		islly 22365	. . . 4 ⊢ ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ((𝐽 ↾t 𝐵) ∈ Top ∧ ∀𝑧 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
3	2	baib 539	. . 3 ⊢ ((𝐽 ↾t 𝐵) ∈ Top → ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑧 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
4	1, 3	syl 17	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑧 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
5		vex 3412	. . . . 5 ⊢ 𝑥 ∈ V
6	5	inex1 5210	. . . 4 ⊢ (𝑥 ∩ 𝐵) ∈ V
7	6	a1i 11	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐵) ∈ V)
8		elrest 16932	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝑧 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐵)))
9		simpr 488	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → 𝑧 = (𝑥 ∩ 𝐵))
10	9	raleqdv 3325	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → (∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
11		rexin 4154	. . . . . 6 ⊢ (∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∃𝑤 ∈ (𝐽 ↾t 𝐵)(𝑤 ∈ 𝒫 𝑧 ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
12		vex 3412	. . . . . . . . 9 ⊢ 𝑢 ∈ V
13	12	inex1 5210	. . . . . . . 8 ⊢ (𝑢 ∩ 𝐵) ∈ V
14	13	a1i 11	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑢 ∈ 𝐽) → (𝑢 ∩ 𝐵) ∈ V)
15		elrest 16932	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝑤 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝑤 = (𝑢 ∩ 𝐵)))
16	15	ad2antrr 726	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (𝑤 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝑤 = (𝑢 ∩ 𝐵)))
17		3anass 1097	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝒫 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ (𝑤 ∈ 𝒫 𝑧 ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)))
18		simpr 488	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑤 = (𝑢 ∩ 𝐵))
19		simpllr 776	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑧 = (𝑥 ∩ 𝐵))
20	18, 19	sseq12d 3934	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑤 ⊆ 𝑧 ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵)))
21		velpw 4518	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝑧 ↔ 𝑤 ⊆ 𝑧)
22		inss2 4144	. . . . . . . . . . . 12 ⊢ (𝑢 ∩ 𝐵) ⊆ 𝐵
23	22	biantru 533	. . . . . . . . . . 11 ⊢ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ↔ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵))
24		ssin 4145	. . . . . . . . . . 11 ⊢ (((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵) ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵))
25	23, 24	bitri 278	. . . . . . . . . 10 ⊢ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵))
26	20, 21, 25	3bitr4g 317	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑤 ∈ 𝒫 𝑧 ↔ (𝑢 ∩ 𝐵) ⊆ 𝑥))
27	18	eleq2d 2823	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ (𝑢 ∩ 𝐵)))
28		simplr 769	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑦 ∈ (𝑥 ∩ 𝐵))
29	28	elin2d 4113	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑦 ∈ 𝐵)
30	29	biantrud 535	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝑢 ∧ 𝑦 ∈ 𝐵)))
31		elin 3882	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑢 ∩ 𝐵) ↔ (𝑦 ∈ 𝑢 ∧ 𝑦 ∈ 𝐵))
32	30, 31	bitr4di 292	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ (𝑢 ∩ 𝐵)))
33	27, 32	bitr4d 285	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑢))
34	18	oveq2d 7229	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → ((𝐽 ↾t 𝐵) ↾t 𝑤) = ((𝐽 ↾t 𝐵) ↾t (𝑢 ∩ 𝐵)))
35		simp-4l 783	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝐽 ∈ Top)
36	22	a1i 11	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑢 ∩ 𝐵) ⊆ 𝐵)
37		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → 𝐵 ∈ 𝑉)
38	37	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝐵 ∈ 𝑉)
39		restabs 22062	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ↾t (𝑢 ∩ 𝐵)) = (𝐽 ↾t (𝑢 ∩ 𝐵)))
40	35, 36, 38, 39	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → ((𝐽 ↾t 𝐵) ↾t (𝑢 ∩ 𝐵)) = (𝐽 ↾t (𝑢 ∩ 𝐵)))
41	34, 40	eqtrd 2777	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → ((𝐽 ↾t 𝐵) ↾t 𝑤) = (𝐽 ↾t (𝑢 ∩ 𝐵)))
42	41	eleq1d 2822	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴 ↔ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴))
43	26, 33, 42	3anbi123d 1438	. . . . . . . 8 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → ((𝑤 ∈ 𝒫 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
44	17, 43	bitr3id 288	. . . . . . 7 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → ((𝑤 ∈ 𝒫 𝑧 ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)) ↔ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
45	14, 16, 44	rexxfr2d 5304	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (∃𝑤 ∈ (𝐽 ↾t 𝐵)(𝑤 ∈ 𝒫 𝑧 ∧ (𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴)) ↔ ∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
46	11, 45	syl5bb 286	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
47	46	ralbidva 3117	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → (∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
48	10, 47	bitrd 282	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → (∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
49	7, 8, 48	ralxfr2d 5303	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (∀𝑧 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑧 ∃𝑤 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑧)(𝑦 ∈ 𝑤 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑤) ∈ 𝐴) ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))
50	4, 49	bitrd 282	1 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ (𝑥 ∩ 𝐵)∃𝑢 ∈ 𝐽 ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t (𝑢 ∩ 𝐵)) ∈ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ∩ 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-bases 21843  df-lly 22363

This theorem is referenced by:  iccllysconn  32925