Theorem kgentopon 23474

Description: The compact generator generates a topology. (Contributed by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
kgentopon	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))

Proof of Theorem kgentopon

Dummy variables 𝑦 𝑥 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniss 4891	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑘Gen‘𝐽) → ∪ 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
2		kgenval 23471	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))})
3		ssrab2 4055	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))} ⊆ 𝒫 𝑋
4	2, 3	eqsstrdi 4003	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝒫 𝑋)
5		sspwuni 5076	. . . . . . . 8 ⊢ ((𝑘Gen‘𝐽) ⊆ 𝒫 𝑋 ↔ ∪ (𝑘Gen‘𝐽) ⊆ 𝑋)
6	4, 5	sylib 218	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ (𝑘Gen‘𝐽) ⊆ 𝑋)
7	1, 6	sylan9ssr 3973	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∪ 𝑥 ⊆ 𝑋)
8		iunin2 5047	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) = (𝑘 ∩ ∪ 𝑦 ∈ 𝑥 𝑦)
9		uniiun 5034	. . . . . . . . . . 11 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
10	9	ineq2i 4192	. . . . . . . . . 10 ⊢ (𝑘 ∩ ∪ 𝑥) = (𝑘 ∩ ∪ 𝑦 ∈ 𝑥 𝑦)
11		incom 4184	. . . . . . . . . 10 ⊢ (𝑘 ∩ ∪ 𝑥) = (∪ 𝑥 ∩ 𝑘)
12	8, 10, 11	3eqtr2i 2764	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) = (∪ 𝑥 ∩ 𝑘)
13		cmptop 23331	. . . . . . . . . . 11 ⊢ ((𝐽 ↾t 𝑘) ∈ Comp → (𝐽 ↾t 𝑘) ∈ Top)
14	13	ad2antll 729	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Top)
15		incom 4184	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ 𝑘) = (𝑘 ∩ 𝑦)
16		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ⊆ (𝑘Gen‘𝐽))
17	16	sselda 3958	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝑘Gen‘𝐽))
18		simplrr 777	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) ∧ 𝑦 ∈ 𝑥) → (𝐽 ↾t 𝑘) ∈ Comp)
19		kgeni 23473	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝑦 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
20	17, 18, 19	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
21	15, 20	eqeltrrid 2839	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) ∧ 𝑦 ∈ 𝑥) → (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘))
22	21	ralrimiva 3132	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ∀𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘))
23		iunopn 22834	. . . . . . . . . 10 ⊢ (((𝐽 ↾t 𝑘) ∈ Top ∧ ∀𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘)) → ∪ 𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘))
24	14, 22, 23	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ∪ 𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘))
25	12, 24	eqeltrrid 2839	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (∪ 𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
26	25	expr 456	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽 ↾t 𝑘) ∈ Comp → (∪ 𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
27	26	ralrimiva 3132	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (∪ 𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
28		elkgen 23472	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∪ 𝑥 ∈ (𝑘Gen‘𝐽) ↔ (∪ 𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (∪ 𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
29	28	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → (∪ 𝑥 ∈ (𝑘Gen‘𝐽) ↔ (∪ 𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (∪ 𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
30	7, 27, 29	mpbir2and 713	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∪ 𝑥 ∈ (𝑘Gen‘𝐽))
31	30	ex 412	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ⊆ (𝑘Gen‘𝐽) → ∪ 𝑥 ∈ (𝑘Gen‘𝐽)))
32	31	alrimiv 1927	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → ∪ 𝑥 ∈ (𝑘Gen‘𝐽)))
33		inss1 4212	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
34		elssuni 4913	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
35	34	ad2antrl 728	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
36		ssidd 3982	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ⊆ 𝑋)
37		elpwi 4582	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
38	37	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 ⊆ 𝑋)
39		sseqin2 4198	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ⊆ 𝑋 ↔ (𝑋 ∩ 𝑘) = 𝑘)
40	38, 39	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑋 ∩ 𝑘) = 𝑘)
41	37	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp) → 𝑘 ⊆ 𝑋)
42		resttopon 23097	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
43	41, 42	sylan2 593	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
44		toponmax 22862	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 ∈ (𝐽 ↾t 𝑘))
45	43, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 ∈ (𝐽 ↾t 𝑘))
46	40, 45	eqeltrd 2834	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑋 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
47	46	expr 456	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽 ↾t 𝑘) ∈ Comp → (𝑋 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
48	47	ralrimiva 3132	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑋 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
49		elkgen 23472	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑋 ∈ (𝑘Gen‘𝐽) ↔ (𝑋 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑋 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
50	36, 48, 49	mpbir2and 713	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ (𝑘Gen‘𝐽))
51		elssuni 4913	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑘Gen‘𝐽) → 𝑋 ⊆ ∪ (𝑘Gen‘𝐽))
52	50, 51	syl 17	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ⊆ ∪ (𝑘Gen‘𝐽))
53	52, 6	eqssd 3976	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ (𝑘Gen‘𝐽))
54	53	adantr 480	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑋 = ∪ (𝑘Gen‘𝐽))
55	35, 54	sseqtrrd 3996	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥 ⊆ 𝑋)
56	33, 55	sstrid 3970	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥 ∩ 𝑦) ⊆ 𝑋)
57		inindir 4211	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝑘) = ((𝑥 ∩ 𝑘) ∩ (𝑦 ∩ 𝑘))
58	13	ad2antll 729	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Top)
59		simplrl 776	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐽))
60		simprr 772	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Comp)
61		kgeni 23473	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
62	59, 60, 61	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
63		simplrr 777	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑦 ∈ (𝑘Gen‘𝐽))
64	63, 60, 19	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑦 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
65		inopn 22835	. . . . . . . . 9 ⊢ (((𝐽 ↾t 𝑘) ∈ Top ∧ (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘) ∧ (𝑦 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) → ((𝑥 ∩ 𝑘) ∩ (𝑦 ∩ 𝑘)) ∈ (𝐽 ↾t 𝑘))
66	58, 62, 64, 65	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((𝑥 ∩ 𝑘) ∩ (𝑦 ∩ 𝑘)) ∈ (𝐽 ↾t 𝑘))
67	57, 66	eqeltrid 2838	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((𝑥 ∩ 𝑦) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
68	67	expr 456	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽 ↾t 𝑘) ∈ Comp → ((𝑥 ∩ 𝑦) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
69	68	ralrimiva 3132	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((𝑥 ∩ 𝑦) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
70		elkgen 23472	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝑥 ∩ 𝑦) ∈ (𝑘Gen‘𝐽) ↔ ((𝑥 ∩ 𝑦) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((𝑥 ∩ 𝑦) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
71	70	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → ((𝑥 ∩ 𝑦) ∈ (𝑘Gen‘𝐽) ↔ ((𝑥 ∩ 𝑦) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((𝑥 ∩ 𝑦) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
72	56, 69, 71	mpbir2and 713	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥 ∩ 𝑦) ∈ (𝑘Gen‘𝐽))
73	72	ralrimivva 3187	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥 ∩ 𝑦) ∈ (𝑘Gen‘𝐽))
74		fvex 6888	. . . 4 ⊢ (𝑘Gen‘𝐽) ∈ V
75		istopg 22831	. . . 4 ⊢ ((𝑘Gen‘𝐽) ∈ V → ((𝑘Gen‘𝐽) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → ∪ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥 ∩ 𝑦) ∈ (𝑘Gen‘𝐽))))
76	74, 75	ax-mp 5	. . 3 ⊢ ((𝑘Gen‘𝐽) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → ∪ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥 ∩ 𝑦) ∈ (𝑘Gen‘𝐽)))
77	32, 73, 76	sylanbrc 583	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ Top)
78		istopon 22848	. 2 ⊢ ((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ↔ ((𝑘Gen‘𝐽) ∈ Top ∧ 𝑋 = ∪ (𝑘Gen‘𝐽)))
79	77, 53, 78	sylanbrc 583	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∀wral 3051  {crab 3415  Vcvv 3459   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575  ∪ cuni 4883  ∪ ciun 4967  ‘cfv 6530  (class class class)co 7403   ↾_t crest 17432  Topctop 22829  TopOnctopon 22846  Compccmp 23322  𝑘Genckgen 23469

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-en 8958  df-fin 8961  df-fi 9421  df-rest 17434  df-topgen 17455  df-top 22830  df-topon 22847  df-bases 22882  df-cmp 23323  df-kgen 23470

This theorem is referenced by:  kgenuni  23475  kgenftop  23476  kgenhaus  23480  kgenidm  23483  kgencn  23492  kgencn3  23494  kgen2cn  23495