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Theorem kgentopon 21669
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.
StepHypRef Expression
1 uniss 4652 . . . . . . 7 (𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
2 kgenval 21666 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})
3 ssrab2 3884 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))} ⊆ 𝒫 𝑋
42, 3syl6eqss 3852 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝒫 𝑋)
5 sspwuni 4803 . . . . . . . 8 ((𝑘Gen‘𝐽) ⊆ 𝒫 𝑋 (𝑘Gen‘𝐽) ⊆ 𝑋)
64, 5sylib 210 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝑋)
71, 6sylan9ssr 3813 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → 𝑥𝑋)
8 iunin2 4775 . . . . . . . . . 10 𝑦𝑥 (𝑘𝑦) = (𝑘 𝑦𝑥 𝑦)
9 uniiun 4764 . . . . . . . . . . 11 𝑥 = 𝑦𝑥 𝑦
109ineq2i 4010 . . . . . . . . . 10 (𝑘 𝑥) = (𝑘 𝑦𝑥 𝑦)
11 incom 4004 . . . . . . . . . 10 (𝑘 𝑥) = ( 𝑥𝑘)
128, 10, 113eqtr2i 2828 . . . . . . . . 9 𝑦𝑥 (𝑘𝑦) = ( 𝑥𝑘)
13 cmptop 21526 . . . . . . . . . . 11 ((𝐽t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Top)
1413ad2antll 721 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
15 incom 4004 . . . . . . . . . . . 12 (𝑦𝑘) = (𝑘𝑦)
16 simplr 786 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ⊆ (𝑘Gen‘𝐽))
1716sselda 3799 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → 𝑦 ∈ (𝑘Gen‘𝐽))
18 simplrr 797 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝐽t 𝑘) ∈ Comp)
19 kgeni 21668 . . . . . . . . . . . . 13 ((𝑦 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑦𝑘) ∈ (𝐽t 𝑘))
2017, 18, 19syl2anc 580 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝑦𝑘) ∈ (𝐽t 𝑘))
2115, 20syl5eqelr 2884 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝑘𝑦) ∈ (𝐽t 𝑘))
2221ralrimiva 3148 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ∀𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
23 iunopn 21030 . . . . . . . . . 10 (((𝐽t 𝑘) ∈ Top ∧ ∀𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘)) → 𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
2414, 22, 23syl2anc 580 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
2512, 24syl5eqelr 2884 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ( 𝑥𝑘) ∈ (𝐽t 𝑘))
2625expr 449 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))
2726ralrimiva 3148 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))
28 elkgen 21667 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → ( 𝑥 ∈ (𝑘Gen‘𝐽) ↔ ( 𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))))
2928adantr 473 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ( 𝑥 ∈ (𝑘Gen‘𝐽) ↔ ( 𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))))
307, 27, 29mpbir2and 705 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → 𝑥 ∈ (𝑘Gen‘𝐽))
3130ex 402 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)))
3231alrimiv 2023 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)))
33 inss1 4029 . . . . . 6 (𝑥𝑦) ⊆ 𝑥
34 elssuni 4660 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
3534ad2antrl 720 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥 (𝑘Gen‘𝐽))
36 ssidd 3821 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝑋)
37 elpwi 4360 . . . . . . . . . . . . . . . 16 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
3837ad2antrl 720 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘𝑋)
39 sseqin2 4016 . . . . . . . . . . . . . . 15 (𝑘𝑋 ↔ (𝑋𝑘) = 𝑘)
4038, 39sylib 210 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑋𝑘) = 𝑘)
4137adantr 473 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp) → 𝑘𝑋)
42 resttopon 21293 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
4341, 42sylan2 587 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
44 toponmax 21058 . . . . . . . . . . . . . . 15 ((𝐽t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 ∈ (𝐽t 𝑘))
4543, 44syl 17 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘 ∈ (𝐽t 𝑘))
4640, 45eqeltrd 2879 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑋𝑘) ∈ (𝐽t 𝑘))
4746expr 449 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))
4847ralrimiva 3148 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))
49 elkgen 21667 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → (𝑋 ∈ (𝑘Gen‘𝐽) ↔ (𝑋𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))))
5036, 48, 49mpbir2and 705 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ (𝑘Gen‘𝐽))
51 elssuni 4660 . . . . . . . . . 10 (𝑋 ∈ (𝑘Gen‘𝐽) → 𝑋 (𝑘Gen‘𝐽))
5250, 51syl 17 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 (𝑘Gen‘𝐽))
5352, 6eqssd 3816 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = (𝑘Gen‘𝐽))
5453adantr 473 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑋 = (𝑘Gen‘𝐽))
5535, 54sseqtr4d 3839 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥𝑋)
5633, 55syl5ss 3810 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥𝑦) ⊆ 𝑋)
57 inindir 4028 . . . . . . . 8 ((𝑥𝑦) ∩ 𝑘) = ((𝑥𝑘) ∩ (𝑦𝑘))
5813ad2antll 721 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
59 simplrl 796 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐽))
60 simprr 790 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Comp)
61 kgeni 21668 . . . . . . . . . 10 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑥𝑘) ∈ (𝐽t 𝑘))
6259, 60, 61syl2anc 580 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ (𝐽t 𝑘))
63 simplrr 797 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑦 ∈ (𝑘Gen‘𝐽))
6463, 60, 19syl2anc 580 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑦𝑘) ∈ (𝐽t 𝑘))
65 inopn 21031 . . . . . . . . 9 (((𝐽t 𝑘) ∈ Top ∧ (𝑥𝑘) ∈ (𝐽t 𝑘) ∧ (𝑦𝑘) ∈ (𝐽t 𝑘)) → ((𝑥𝑘) ∩ (𝑦𝑘)) ∈ (𝐽t 𝑘))
6658, 62, 64, 65syl3anc 1491 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑥𝑘) ∩ (𝑦𝑘)) ∈ (𝐽t 𝑘))
6757, 66syl5eqel 2883 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘))
6867expr 449 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))
6968ralrimiva 3148 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))
70 elkgen 21667 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → ((𝑥𝑦) ∈ (𝑘Gen‘𝐽) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
7170adantr 473 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → ((𝑥𝑦) ∈ (𝑘Gen‘𝐽) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
7256, 69, 71mpbir2and 705 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥𝑦) ∈ (𝑘Gen‘𝐽))
7372ralrimivva 3153 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥𝑦) ∈ (𝑘Gen‘𝐽))
74 fvex 6425 . . . 4 (𝑘Gen‘𝐽) ∈ V
75 istopg 21027 . . . 4 ((𝑘Gen‘𝐽) ∈ V → ((𝑘Gen‘𝐽) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥𝑦) ∈ (𝑘Gen‘𝐽))))
7674, 75ax-mp 5 . . 3 ((𝑘Gen‘𝐽) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥𝑦) ∈ (𝑘Gen‘𝐽)))
7732, 73, 76sylanbrc 579 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ Top)
78 istopon 21044 . 2 ((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ↔ ((𝑘Gen‘𝐽) ∈ Top ∧ 𝑋 = (𝑘Gen‘𝐽)))
7977, 53, 78sylanbrc 579 1 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651   = wceq 1653  wcel 2157  wral 3090  {crab 3094  Vcvv 3386  cin 3769  wss 3770  𝒫 cpw 4350   cuni 4629   ciun 4711  cfv 6102  (class class class)co 6879  t crest 16395  Topctop 21025  TopOnctopon 21042  Compccmp 21517  𝑘Genckgen 21664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-om 7301  df-1st 7402  df-2nd 7403  df-wrecs 7646  df-recs 7708  df-rdg 7746  df-oadd 7804  df-er 7983  df-en 8197  df-fin 8200  df-fi 8560  df-rest 16397  df-topgen 16418  df-top 21026  df-topon 21043  df-bases 21078  df-cmp 21518  df-kgen 21665
This theorem is referenced by:  kgenuni  21670  kgenftop  21671  kgenhaus  21675  kgenidm  21678  kgencn  21687  kgencn3  21689  kgen2cn  21690
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