Theorem kgentopon 23664

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 4884	. . . . . . 7 ⊢ (𝑥 ⊆ (𝑘Gen‘𝐽) → ∪ 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
2		kgenval 23661	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))})
3		ssrab2 4042	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))} ⊆ 𝒫 𝑋
4	2, 3	eqsstrdi 3989	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝒫 𝑋)
5		sspwuni 5070	. . . . . . . 8 ⊢ ((𝑘Gen‘𝐽) ⊆ 𝒫 𝑋 ↔ ∪ (𝑘Gen‘𝐽) ⊆ 𝑋)
6	4, 5	sylib 221	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ (𝑘Gen‘𝐽) ⊆ 𝑋)
7	1, 6	sylan9ssr 3959	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∪ 𝑥 ⊆ 𝑋)
8		iunin2 5039	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) = (𝑘 ∩ ∪ 𝑦 ∈ 𝑥 𝑦)
9		uniiun 5027	. . . . . . . . . . 11 ⊢ ∪ 𝑥 = ∪ 𝑦 ∈ 𝑥 𝑦
10	9	ineq2i 4178	. . . . . . . . . 10 ⊢ (𝑘 ∩ ∪ 𝑥) = (𝑘 ∩ ∪ 𝑦 ∈ 𝑥 𝑦)
11		incom 4170	. . . . . . . . . 10 ⊢ (𝑘 ∩ ∪ 𝑥) = (∪ 𝑥 ∩ 𝑘)
12	8, 10, 11	3eqtr2i 2798	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) = (∪ 𝑥 ∩ 𝑘)
13		cmptop 23521	. . . . . . . . . . 11 ⊢ ((𝐽 ↾t 𝑘) ∈ Comp → (𝐽 ↾t 𝑘) ∈ Top)
14	13	ad2antll 741	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Top)
15		incom 4170	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ 𝑘) = (𝑘 ∩ 𝑦)
16		simplr 780	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ⊆ (𝑘Gen‘𝐽))
17	16	sselda 3945	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝑘Gen‘𝐽))
18		simplrr 789	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) ∧ 𝑦 ∈ 𝑥) → (𝐽 ↾t 𝑘) ∈ Comp)
19		kgeni 23663	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝑦 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
20	17, 18, 19	syl2anc 595	. . . . . . . . . . . 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 2874	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) ∧ 𝑦 ∈ 𝑥) → (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘))
22	21	ralrimiva 3163	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ∀𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘))
23		iunopn 23024	. . . . . . . . . 10 ⊢ (((𝐽 ↾t 𝑘) ∈ Top ∧ ∀𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘)) → ∪ 𝑦 ∈ 𝑥 (𝑘 ∩ 𝑦) ∈ (𝐽 ↾t 𝑘))
24	14, 22, 23	syl2anc 595	. . . . . . . . 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 2874	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (∪ 𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
26	25	expr 461	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽 ↾t 𝑘) ∈ Comp → (∪ 𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
27	26	ralrimiva 3163	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (∪ 𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
28		elkgen 23662	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∪ 𝑥 ∈ (𝑘Gen‘𝐽) ↔ (∪ 𝑥 ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (∪ 𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))))
29	28	adantr 485	. . . . . 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 725	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∪ 𝑥 ∈ (𝑘Gen‘𝐽))
31	30	ex 417	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ⊆ (𝑘Gen‘𝐽) → ∪ 𝑥 ∈ (𝑘Gen‘𝐽)))
32	31	alrimiv 1954	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → ∪ 𝑥 ∈ (𝑘Gen‘𝐽)))
33		inss1 4197	. . . . . 6 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥
34		elssuni 4908	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
35	34	ad2antrl 740	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
36		ssidd 3968	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ⊆ 𝑋)
37		elpwi 4574	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋)
38	37	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 ⊆ 𝑋)
39		sseqin2 4184	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ⊆ 𝑋 ↔ (𝑋 ∩ 𝑘) = 𝑘)
40	38, 39	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑋 ∩ 𝑘) = 𝑘)
41	37	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp) → 𝑘 ⊆ 𝑋)
42		resttopon 23287	. . . . . . . . . . . . . . . 16 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
43	41, 42	sylan2 604	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘))
44		toponmax 23052	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 ∈ (𝐽 ↾t 𝑘))
45	43, 44	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 ∈ (𝐽 ↾t 𝑘))
46	40, 45	eqeltrd 2869	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑋 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
47	46	expr 461	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽 ↾t 𝑘) ∈ Comp → (𝑋 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
48	47	ralrimiva 3163	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝑋 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))
49		elkgen 23662	. . . . . . . . . . 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 725	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ (𝑘Gen‘𝐽))
51		elssuni 4908	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝑘Gen‘𝐽) → 𝑋 ⊆ ∪ (𝑘Gen‘𝐽))
52	50, 51	syl 18	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ⊆ ∪ (𝑘Gen‘𝐽))
53	52, 6	eqssd 3962	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ (𝑘Gen‘𝐽))
54	53	adantr 485	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑋 = ∪ (𝑘Gen‘𝐽))
55	35, 54	sseqtrrd 3982	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥 ⊆ 𝑋)
56	33, 55	sstrid 3956	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥 ∩ 𝑦) ⊆ 𝑋)
57		inindir 4196	. . . . . . . 8 ⊢ ((𝑥 ∩ 𝑦) ∩ 𝑘) = ((𝑥 ∩ 𝑘) ∩ (𝑦 ∩ 𝑘))
58	13	ad2antll 741	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Top)
59		simplrl 788	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐽))
60		simprr 784	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Comp)
61		kgeni 23663	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝑘) ∈ Comp) → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
62	59, 60, 61	syl2anc 595	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑥 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
63		simplrr 789	. . . . . . . . . 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 595	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑦 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))
65		inopn 23025	. . . . . . . . 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 1396	. . . . . . . 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 2873	. . . . . . 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 461	. . . . . 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 3163	. . . . 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 23662	. . . . . 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 485	. . . . 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 725	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥 ∩ 𝑦) ∈ (𝑘Gen‘𝐽))
73	72	ralrimivva 3214	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥 ∩ 𝑦) ∈ (𝑘Gen‘𝐽))
74		fvex 6895	. . . 4 ⊢ (𝑘Gen‘𝐽) ∈ V
75		istopg 23021	. . . 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 594	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ Top)
78		istopon 23038	. 2 ⊢ ((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ↔ ((𝑘Gen‘𝐽) ∈ Top ∧ 𝑋 = ∪ (𝑘Gen‘𝐽)))
79	77, 53, 78	sylanbrc 594	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4876  ∪ ciun 4960  ‘cfv 6537  (class class class)co 7411   ↾_t crest 17473  Topctop 23019  TopOnctopon 23036  Compccmp 23512  𝑘Genckgen 23659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-en 8944  df-fin 8947  df-fi 9371  df-rest 17475  df-topgen 17496  df-top 23020  df-topon 23037  df-bases 23072  df-cmp 23513  df-kgen 23660

This theorem is referenced by:  kgenuni  23665  kgenftop  23666  kgenhaus  23670  kgenidm  23673  kgencn  23682  kgencn3  23684  kgen2cn  23685