Theorem subislly 22540

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 22219	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝐽 ↾t 𝐵) ∈ Top)
2		islly 22527	. . . 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 535	. . 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 3426	. . . . 5 ⊢ 𝑥 ∈ V
6	5	inex1 5236	. . . 4 ⊢ (𝑥 ∩ 𝐵) ∈ V
7	6	a1i 11	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐵) ∈ V)
8		elrest 17055	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝑧 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐵)))
9		simpr 484	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → 𝑧 = (𝑥 ∩ 𝐵))
10	9	raleqdv 3339	. . . 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 4170	. . . . . 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 3426	. . . . . . . . 9 ⊢ 𝑢 ∈ V
13	12	inex1 5236	. . . . . . . 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 17055	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝑤 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝑤 = (𝑢 ∩ 𝐵)))
16	15	ad2antrr 722	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (𝑤 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝑤 = (𝑢 ∩ 𝐵)))
17		3anass 1093	. . . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑤 = (𝑢 ∩ 𝐵))
19		simpllr 772	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑧 = (𝑥 ∩ 𝐵))
20	18, 19	sseq12d 3950	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑤 ⊆ 𝑧 ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵)))
21		velpw 4535	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝑧 ↔ 𝑤 ⊆ 𝑧)
22		inss2 4160	. . . . . . . . . . . 12 ⊢ (𝑢 ∩ 𝐵) ⊆ 𝐵
23	22	biantru 529	. . . . . . . . . . 11 ⊢ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ↔ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵))
24		ssin 4161	. . . . . . . . . . 11 ⊢ (((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵) ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵))
25	23, 24	bitri 274	. . . . . . . . . 10 ⊢ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵))
26	20, 21, 25	3bitr4g 313	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑤 ∈ 𝒫 𝑧 ↔ (𝑢 ∩ 𝐵) ⊆ 𝑥))
27	18	eleq2d 2824	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ (𝑢 ∩ 𝐵)))
28		simplr 765	. . . . . . . . . . . . 13 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑦 ∈ (𝑥 ∩ 𝐵))
29	28	elin2d 4129	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑦 ∈ 𝐵)
30	29	biantrud 531	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝑢 ∧ 𝑦 ∈ 𝐵)))
31		elin 3899	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑢 ∩ 𝐵) ↔ (𝑦 ∈ 𝑢 ∧ 𝑦 ∈ 𝐵))
32	30, 31	bitr4di 288	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ (𝑢 ∩ 𝐵)))
33	27, 32	bitr4d 281	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑢))
34	18	oveq2d 7271	. . . . . . . . . . 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 779	. . . . . . . . . . . 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 765	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → 𝐵 ∈ 𝑉)
38	37	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝐵 ∈ 𝑉)
39		restabs 22224	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ↾t (𝑢 ∩ 𝐵)) = (𝐽 ↾t (𝑢 ∩ 𝐵)))
40	35, 36, 38, 39	syl3anc 1369	. . . . . . . . . . 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 2778	. . . . . . . . . 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 2823	. . . . . . . . 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 1434	. . . . . . . 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 284	. . . . . . 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 5329	. . . . . 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 282	. . . . 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 3119	. . . 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 278	. . 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 5328	. 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 278	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  (class class class)co 7255   ↾_t crest 17048  Topctop 21950  Locally clly 22523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-en 8692  df-fin 8695  df-fi 9100  df-rest 17050  df-topgen 17071  df-top 21951  df-bases 22004  df-lly 22525

This theorem is referenced by:  iccllysconn  33112