Theorem subislly 23437

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 23116	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝐽 ↾t 𝐵) ∈ Top)
2		islly 23424	. . . 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 3446	. . . . 5 ⊢ 𝑥 ∈ V
6	5	inex1 5264	. . . 4 ⊢ (𝑥 ∩ 𝐵) ∈ V
7	6	a1i 11	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐵) ∈ V)
8		elrest 17359	. . 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 3298	. . . 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 4204	. . . . . 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 3446	. . . . . . . . 9 ⊢ 𝑢 ∈ V
13	12	inex1 5264	. . . . . . . 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 17359	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝑤 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝑤 = (𝑢 ∩ 𝐵)))
16	15	ad2antrr 727	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) → (𝑤 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝑤 = (𝑢 ∩ 𝐵)))
17		3anass 1095	. . . . . . . 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 776	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑧 = (𝑥 ∩ 𝐵))
20	18, 19	sseq12d 3969	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑤 ⊆ 𝑧 ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵)))
21		velpw 4561	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝑧 ↔ 𝑤 ⊆ 𝑧)
22		inss2 4192	. . . . . . . . . . . 12 ⊢ (𝑢 ∩ 𝐵) ⊆ 𝐵
23	22	biantru 529	. . . . . . . . . . 11 ⊢ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ↔ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵))
24		ssin 4193	. . . . . . . . . . 11 ⊢ (((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵) ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵))
25	23, 24	bitri 275	. . . . . . . . . 10 ⊢ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵))
26	20, 21, 25	3bitr4g 314	. . . . . . . . 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 4159	. . . . . . . . . . . 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 3919	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑢 ∩ 𝐵) ↔ (𝑦 ∈ 𝑢 ∧ 𝑦 ∈ 𝐵))
32	30, 31	bitr4di 289	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ (𝑢 ∩ 𝐵)))
33	27, 32	bitr4d 282	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑢))
34	18	oveq2d 7384	. . . . . . . . . . 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 727	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝐵 ∈ 𝑉)
39		restabs 23121	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ↾t (𝑢 ∩ 𝐵)) = (𝐽 ↾t (𝑢 ∩ 𝐵)))
40	35, 36, 38, 39	syl3anc 1374	. . . . . . . . . . 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 2772	. . . . . . . . . 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 1439	. . . . . . . 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 285	. . . . . . 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 5358	. . . . . 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	bitrid 283	. . . . 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 3159	. . . 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 279	. . 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 5357	. 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 279	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  (class class class)co 7368   ↾_t crest 17352  Topctop 22849  Locally clly 23420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-en 8896  df-fin 8899  df-fi 9326  df-rest 17354  df-topgen 17375  df-top 22850  df-bases 22902  df-lly 23422

This theorem is referenced by:  iccllysconn  35463