Theorem kgentopon 23503

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 4858	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑘Gen‘𝐽) → ∪ 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
2		kgenval 23500	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))})
3		ssrab2 4020	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))} ⊆ 𝒫 𝑋
4	2, 3	eqsstrdi 3966	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝒫 𝑋)
5		sspwuni 5042	. . . . . . . 8 ⊢ ((𝑘Gen‘𝐽) ⊆ 𝒫 𝑋 ↔ ∪ (𝑘Gen‘𝐽) ⊆ 𝑋)
6	4, 5	sylib 218	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ (𝑘Gen‘𝐽) ⊆ 𝑋)
7	1, 6	sylan9ssr 3936	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∪ 𝑥 ⊆ 𝑋)
8		iunin2 5013	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) = (𝑘 ∩ ∪ 𝑦 ∈ 𝑥 𝑦)
9		uniiun 5001	. . . . . . . . . . 11 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
10	9	ineq2i 4157	. . . . . . . . . 10 ⊢ (𝑘 ∩ ∪ 𝑥) = (𝑘 ∩ ∪ 𝑦 ∈ 𝑥 𝑦)
11		incom 4149	. . . . . . . . . 10 ⊢ (𝑘 ∩ ∪ 𝑥) = (∪ 𝑥 ∩ 𝑘)
12	8, 10, 11	3eqtr2i 2765	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) = (∪ 𝑥 ∩ 𝑘)
13		cmptop 23360	. . . . . . . . . . 11 ⊢ ((𝐽 ↾t 𝑘) ∈ Comp → (𝐽 ↾t 𝑘) ∈ Top)
14	13	ad2antll 730	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Top)
15		incom 4149	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ 𝑘) = (𝑘 ∩ 𝑦)
16		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ⊆ (𝑘Gen‘𝐽))
17	16	sselda 3921	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝑘Gen‘𝐽))
18		simplrr 778	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) ∧ 𝑦 ∈ 𝑥) → (𝐽 ↾t 𝑘) ∈ Comp)
19		kgeni 23502	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝑦 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
20	17, 18, 19	syl2anc 585	. . . . . . . . . . . 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 2841	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) ∧ 𝑦 ∈ 𝑥) → (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘))
22	21	ralrimiva 3129	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ∀𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘))
23		iunopn 22863	. . . . . . . . . 10 ⊢ (((𝐽 ↾t 𝑘) ∈ Top ∧ ∀𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘)) → ∪ 𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘))
24	14, 22, 23	syl2anc 585	. . . . . . . . 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 2841	. . . . . . . 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 3129	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (∪ 𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
28		elkgen 23501	. . . . . . 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 714	. . . . 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 1929	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → ∪ 𝑥 ∈ (𝑘Gen‘𝐽)))
33		inss1 4177	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
34		elssuni 4881	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
35	34	ad2antrl 729	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
36		ssidd 3945	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ⊆ 𝑋)
37		elpwi 4548	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
38	37	ad2antrl 729	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 ⊆ 𝑋)
39		sseqin2 4163	. . . . . . . . . . . . . . 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 23126	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
43	41, 42	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
44		toponmax 22891	. . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . 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 3129	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑋 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
49		elkgen 23501	. . . . . . . . . . 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 714	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ (𝑘Gen‘𝐽))
51		elssuni 4881	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑘Gen‘𝐽) → 𝑋 ⊆ ∪ (𝑘Gen‘𝐽))
52	50, 51	syl 17	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ⊆ ∪ (𝑘Gen‘𝐽))
53	52, 6	eqssd 3939	. . . . . . . 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 3959	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥 ⊆ 𝑋)
56	33, 55	sstrid 3933	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥 ∩ 𝑦) ⊆ 𝑋)
57		inindir 4176	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝑘) = ((𝑥 ∩ 𝑘) ∩ (𝑦 ∩ 𝑘))
58	13	ad2antll 730	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Top)
59		simplrl 777	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐽))
60		simprr 773	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Comp)
61		kgeni 23502	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
62	59, 60, 61	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
63		simplrr 778	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑦 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
65		inopn 22864	. . . . . . . . 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 1374	. . . . . . . 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 2840	. . . . . . 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 3129	. . . . 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 23501	. . . . . 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 714	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥 ∩ 𝑦) ∈ (𝑘Gen‘𝐽))
73	72	ralrimivva 3180	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥 ∩ 𝑦) ∈ (𝑘Gen‘𝐽))
74		fvex 6853	. . . 4 ⊢ (𝑘Gen‘𝐽) ∈ V
75		istopg 22860	. . . 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 584	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ Top)
78		istopon 22877	. 2 ⊢ ((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ↔ ((𝑘Gen‘𝐽) ∈ Top ∧ 𝑋 = ∪ (𝑘Gen‘𝐽)))
79	77, 53, 78	sylanbrc 584	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850  ∪ ciun 4933  ‘cfv 6498  (class class class)co 7367   ↾_t crest 17383  Topctop 22858  TopOnctopon 22875  Compccmp 23351  𝑘Genckgen 23498

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-en 8894  df-fin 8897  df-fi 9324  df-rest 17385  df-topgen 17406  df-top 22859  df-topon 22876  df-bases 22911  df-cmp 23352  df-kgen 23499

This theorem is referenced by:  kgenuni  23504  kgenftop  23505  kgenhaus  23509  kgenidm  23512  kgencn  23521  kgencn3  23523  kgen2cn  23524