Theorem subislly 21773

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 21452	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝐽 ↾t 𝐵) ∈ Top)
2		islly 21760	. . . 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 536	. . 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 3440	. . . . 5 ⊢ 𝑥 ∈ V
6	5	inex1 5112	. . . 4 ⊢ (𝑥 ∩ 𝐵) ∈ V
7	6	a1i 11	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐵) ∈ V)
8		elrest 16530	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) → (𝑧 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐵)))
9		simpr 485	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → 𝑧 = (𝑥 ∩ 𝐵))
10	9	raleqdv 3375	. . . 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 4136	. . . . . 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 3440	. . . . . . . . 9 ⊢ 𝑢 ∈ V
13	12	inex1 5112	. . . . . . . 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 16530	. . . . . . . 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 1088	. . . . . . . 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 485	. . . . . . . . . . 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 3921	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑤 ⊆ 𝑧 ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵)))
21		selpw 4460	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝑧 ↔ 𝑤 ⊆ 𝑧)
22		inss2 4126	. . . . . . . . . . . 12 ⊢ (𝑢 ∩ 𝐵) ⊆ 𝐵
23	22	biantru 530	. . . . . . . . . . 11 ⊢ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ↔ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵))
24		ssin 4127	. . . . . . . . . . 11 ⊢ (((𝑢 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵) ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵))
25	23, 24	bitri 276	. . . . . . . . . 10 ⊢ ((𝑢 ∩ 𝐵) ⊆ 𝑥 ↔ (𝑢 ∩ 𝐵) ⊆ (𝑥 ∩ 𝐵))
26	20, 21, 25	3bitr4g 315	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑤 ∈ 𝒫 𝑧 ↔ (𝑢 ∩ 𝐵) ⊆ 𝑥))
27	18	eleq2d 2868	. . . . . . . . . 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 4097	. . . . . . . . . . . 12 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → 𝑦 ∈ 𝐵)
30	29	biantrud 532	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑢 ↔ (𝑦 ∈ 𝑢 ∧ 𝑦 ∈ 𝐵)))
31		elin 4090	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑢 ∩ 𝐵) ↔ (𝑦 ∈ 𝑢 ∧ 𝑦 ∈ 𝐵))
32	30, 31	syl6bbr 290	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ (𝑢 ∩ 𝐵)))
33	27, 32	bitr4d 283	. . . . . . . . 9 ⊢ (((((𝐽 ∈ Top ∧ 𝐵 ∈ 𝑉) ∧ 𝑧 = (𝑥 ∩ 𝐵)) ∧ 𝑦 ∈ (𝑥 ∩ 𝐵)) ∧ 𝑤 = (𝑢 ∩ 𝐵)) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ 𝑢))
34	18	oveq2d 7032	. . . . . . . . . . 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 21457	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∩ 𝐵) ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → ((𝐽 ↾t 𝐵) ↾t (𝑢 ∩ 𝐵)) = (𝐽 ↾t (𝑢 ∩ 𝐵)))
40	35, 36, 38, 39	syl3anc 1364	. . . . . . . . . . 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 2831	. . . . . . . . . 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 2867	. . . . . . . . 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 1428	. . . . . . . 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	syl5bbr 286	. . . . . . 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 5203	. . . . . 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 284	. . . . 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 3163	. . . 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 280	. . 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 5202	. 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 280	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 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  Vcvv 3437   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4453  (class class class)co 7016   ↾_t crest 16523  Topctop 21185  Locally clly 21756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-oadd 7957  df-er 8139  df-en 8358  df-fin 8361  df-fi 8721  df-rest 16525  df-topgen 16546  df-top 21186  df-bases 21238  df-lly 21758

This theorem is referenced by:  iccllysconn  32105