Theorem llycmpkgen2 23558

Description: A locally compact space is compactly generated. (This variant of llycmpkgen 23560 uses the weaker definition of locally compact, "every point has a compact neighborhood", instead of "every point has a local base of compact neighborhoods".) (Contributed by Mario Carneiro, 21-Mar-2015.)

Hypotheses

Ref	Expression
iskgen3.1	⊢ 𝑋 = ∪ 𝐽
llycmpkgen2.2	⊢ (𝜑 → 𝐽 ∈ Top)
llycmpkgen2.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp)

Assertion

Ref	Expression
llycmpkgen2	⊢ (𝜑 → 𝐽 ∈ ran 𝑘Gen)

Distinct variable groups:   𝑥,𝑘,𝐽   𝜑,𝑘,𝑥   𝑘,𝑋

Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem llycmpkgen2

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

Step	Hyp	Ref	Expression
1		llycmpkgen2.2	. 2 ⊢ (𝜑 → 𝐽 ∈ Top)
2		elssuni 4937	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢 ⊆ ∪ (𝑘Gen‘𝐽))
3	2	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢 ⊆ ∪ (𝑘Gen‘𝐽))
4		iskgen3.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
5	4	kgenuni 23547	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ Top → 𝑋 = ∪ (𝑘Gen‘𝐽))
6	1, 5	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ∪ (𝑘Gen‘𝐽))
7	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑋 = ∪ (𝑘Gen‘𝐽))
8	3, 7	sseqtrrd 4021	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢 ⊆ 𝑋)
9	8	sselda 3983	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝑋)
10		llycmpkgen2.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp)
11	10	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp)
12	9, 11	syldan 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp)
13	1	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝐽 ∈ Top)
14		difss 4136	. . . . . . . . . 10 ⊢ (𝑋 ∖ (𝑘 ∖ 𝑢)) ⊆ 𝑋
15	4	ntropn 23057	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘 ∖ 𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∈ 𝐽)
16	13, 14, 15	sylancl 586	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∈ 𝐽)
17		simprl 771	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 ∈ ((nei‘𝐽)‘{𝑥}))
18	4	neii1 23114	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑘 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑘 ⊆ 𝑋)
19	13, 17, 18	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 ⊆ 𝑋)
20	4	ntropn 23057	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑘 ⊆ 𝑋) → ((int‘𝐽)‘𝑘) ∈ 𝐽)
21	13, 19, 20	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ∈ 𝐽)
22		inopn 22905	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∈ 𝐽 ∧ ((int‘𝐽)‘𝑘) ∈ 𝐽) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽)
23	13, 16, 21, 22	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽)
24		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ 𝑢)
25	4	ntrss2 23065	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ 𝑘 ⊆ 𝑋) → ((int‘𝐽)‘𝑘) ⊆ 𝑘)
26	13, 19, 25	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ⊆ 𝑘)
27	9	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ 𝑋)
28	27	snssd 4809	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → {𝑥} ⊆ 𝑋)
29	4	neiint 23112	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝑋 ∧ 𝑘 ⊆ 𝑋) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘)))
30	13, 28, 19, 29	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘)))
31	17, 30	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → {𝑥} ⊆ ((int‘𝐽)‘𝑘))
32		vex 3484	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
33	32	snss 4785	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ((int‘𝐽)‘𝑘) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘))
34	31, 33	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘𝑘))
35	26, 34	sseldd 3984	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ 𝑘)
36	24, 35	elind 4200	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑢 ∩ 𝑘))
37		simpllr 776	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑢 ∈ (𝑘Gen‘𝐽))
38		simprr 773	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Comp)
39		kgeni 23545	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝑢 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
40	37, 38, 39	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑢 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
41		vex 3484	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
42		resttop 23168	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ Top ∧ 𝑘 ∈ V) → (𝐽 ↾t 𝑘) ∈ Top)
43	13, 41, 42	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Top)
44		inss2 4238	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∩ 𝑘) ⊆ 𝑘
45	4	restuni 23170	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑘 ⊆ 𝑋) → 𝑘 = ∪ (𝐽 ↾t 𝑘))
46	13, 19, 45	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 = ∪ (𝐽 ↾t 𝑘))
47	44, 46	sseqtrid 4026	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑢 ∩ 𝑘) ⊆ ∪ (𝐽 ↾t 𝑘))
48		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ ∪ (𝐽 ↾t 𝑘) = ∪ (𝐽 ↾t 𝑘)
49	48	isopn3 23074	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ↾t 𝑘) ∈ Top ∧ (𝑢 ∩ 𝑘) ⊆ ∪ (𝐽 ↾t 𝑘)) → ((𝑢 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘) ↔ ((int‘(𝐽 ↾t 𝑘))‘(𝑢 ∩ 𝑘)) = (𝑢 ∩ 𝑘)))
50	43, 47, 49	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((𝑢 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘) ↔ ((int‘(𝐽 ↾t 𝑘))‘(𝑢 ∩ 𝑘)) = (𝑢 ∩ 𝑘)))
51	40, 50	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘(𝐽 ↾t 𝑘))‘(𝑢 ∩ 𝑘)) = (𝑢 ∩ 𝑘))
52	44	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑢 ∩ 𝑘) ⊆ 𝑘)
53		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝑘)
54	4, 53	restntr 23190	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑘 ⊆ 𝑋 ∧ (𝑢 ∩ 𝑘) ⊆ 𝑘) → ((int‘(𝐽 ↾t 𝑘))‘(𝑢 ∩ 𝑘)) = (((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) ∩ 𝑘))
55	13, 19, 52, 54	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘(𝐽 ↾t 𝑘))‘(𝑢 ∩ 𝑘)) = (((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) ∩ 𝑘))
56	51, 55	eqtr3d 2779	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑢 ∩ 𝑘) = (((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) ∩ 𝑘))
57	36, 56	eleqtrd 2843	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) ∩ 𝑘))
58	57	elin1d 4204	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))))
59		undif3 4300	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘)) = (((𝑢 ∩ 𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢 ∩ 𝑘)))
60		incom 4209	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∩ 𝑘) = (𝑘 ∩ 𝑢)
61	60	difeq2i 4123	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∖ (𝑢 ∩ 𝑘)) = (𝑘 ∖ (𝑘 ∩ 𝑢))
62		difin 4272	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∖ (𝑘 ∩ 𝑢)) = (𝑘 ∖ 𝑢)
63	61, 62	eqtri 2765	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∖ (𝑢 ∩ 𝑘)) = (𝑘 ∖ 𝑢)
64	63	difeq2i 4123	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∩ 𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢 ∩ 𝑘))) = (((𝑢 ∩ 𝑘) ∪ 𝑋) ∖ (𝑘 ∖ 𝑢))
65	59, 64	eqtri 2765	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘)) = (((𝑢 ∩ 𝑘) ∪ 𝑋) ∖ (𝑘 ∖ 𝑢))
66	44, 19	sstrid 3995	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑢 ∩ 𝑘) ⊆ 𝑋)
67		ssequn1 4186	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∩ 𝑘) ⊆ 𝑋 ↔ ((𝑢 ∩ 𝑘) ∪ 𝑋) = 𝑋)
68	66, 67	sylib 218	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((𝑢 ∩ 𝑘) ∪ 𝑋) = 𝑋)
69	68	difeq1d 4125	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((𝑢 ∩ 𝑘) ∪ 𝑋) ∖ (𝑘 ∖ 𝑢)) = (𝑋 ∖ (𝑘 ∖ 𝑢)))
70	65, 69	eqtrid 2789	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘)) = (𝑋 ∖ (𝑘 ∖ 𝑢)))
71	70	fveq2d 6910	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) = ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))))
72	58, 71	eleqtrd 2843	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))))
73	72, 34	elind 4200	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)))
74		sslin 4243	. . . . . . . . . 10 ⊢ (((int‘𝐽)‘𝑘) ⊆ 𝑘 → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ 𝑘))
75	26, 74	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ 𝑘))
76	4	ntrss2 23065	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘 ∖ 𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ (𝑋 ∖ (𝑘 ∖ 𝑢)))
77	13, 14, 76	sylancl 586	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ (𝑋 ∖ (𝑘 ∖ 𝑢)))
78	77	difss2d 4139	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ 𝑋)
79		reldisj 4453	. . . . . . . . . . . 12 ⊢ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ 𝑋 → ((((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ (𝑘 ∖ 𝑢)) = ∅ ↔ ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ (𝑋 ∖ (𝑘 ∖ 𝑢))))
80	78, 79	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ (𝑘 ∖ 𝑢)) = ∅ ↔ ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ (𝑋 ∖ (𝑘 ∖ 𝑢))))
81	77, 80	mpbird 257	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ (𝑘 ∖ 𝑢)) = ∅)
82		inssdif0 4374	. . . . . . . . . 10 ⊢ ((((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ 𝑘) ⊆ 𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ (𝑘 ∖ 𝑢)) = ∅)
83	81, 82	sylibr 234	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ 𝑘) ⊆ 𝑢)
84	75, 83	sstrd 3994	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)
85		eleq2 2830	. . . . . . . . . 10 ⊢ (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘))))
86		sseq1 4009	. . . . . . . . . 10 ⊢ (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑧 ⊆ 𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢))
87	85, 86	anbi12d 632	. . . . . . . . 9 ⊢ (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) → ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ↔ (𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∧ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)))
88	87	rspcev 3622	. . . . . . . 8 ⊢ (((((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽 ∧ (𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∧ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))
89	23, 73, 84, 88	syl12anc 837	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))
90	12, 89	rexlimddv 3161	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))
91	90	ralrimiva 3146	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) → ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))
92	91	ex 412	. . . 4 ⊢ (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)))
93		eltop2 22982	. . . . 5 ⊢ (𝐽 ∈ Top → (𝑢 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)))
94	1, 93	syl 17	. . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)))
95	92, 94	sylibrd 259	. . 3 ⊢ (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢 ∈ 𝐽))
96	95	ssrdv 3989	. 2 ⊢ (𝜑 → (𝑘Gen‘𝐽) ⊆ 𝐽)
97		iskgen2 23556	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
98	1, 96, 97	sylanbrc 583	1 ⊢ (𝜑 → 𝐽 ∈ ran 𝑘Gen)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626  ∪ cuni 4907  ran crn 5686  ‘cfv 6561  (class class class)co 7431   ↾_t crest 17465  Topctop 22899  intcnt 23025  neicnei 23105  Compccmp 23394  𝑘Genckgen 23541

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-ntr 23028  df-nei 23106  df-cmp 23395  df-kgen 23542

This theorem is referenced by:  cmpkgen  23559  llycmpkgen  23560