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Theorem kgentopon 21151
 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 4394 . . . . . . 7 (𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
2 kgenval 21148 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})
3 ssrab2 3650 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))} ⊆ 𝒫 𝑋
42, 3syl6eqss 3618 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝒫 𝑋)
5 sspwuni 4547 . . . . . . . 8 ((𝑘Gen‘𝐽) ⊆ 𝒫 𝑋 (𝑘Gen‘𝐽) ⊆ 𝑋)
64, 5sylib 207 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝑋)
71, 6sylan9ssr 3582 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → 𝑥𝑋)
8 iunin2 4520 . . . . . . . . . 10 𝑦𝑥 (𝑘𝑦) = (𝑘 𝑦𝑥 𝑦)
9 uniiun 4509 . . . . . . . . . . 11 𝑥 = 𝑦𝑥 𝑦
109ineq2i 3773 . . . . . . . . . 10 (𝑘 𝑥) = (𝑘 𝑦𝑥 𝑦)
11 incom 3767 . . . . . . . . . 10 (𝑘 𝑥) = ( 𝑥𝑘)
128, 10, 113eqtr2i 2638 . . . . . . . . 9 𝑦𝑥 (𝑘𝑦) = ( 𝑥𝑘)
13 cmptop 21008 . . . . . . . . . . 11 ((𝐽t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Top)
1413ad2antll 761 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
15 incom 3767 . . . . . . . . . . . 12 (𝑦𝑘) = (𝑘𝑦)
16 simplr 788 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ⊆ (𝑘Gen‘𝐽))
1716sselda 3568 . . . . . . . . . . . . 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 21150 . . . . . . . . . . . . 13 ((𝑦 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑦𝑘) ∈ (𝐽t 𝑘))
2017, 18, 19syl2anc 691 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝑦𝑘) ∈ (𝐽t 𝑘))
2115, 20syl5eqelr 2693 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝑘𝑦) ∈ (𝐽t 𝑘))
2221ralrimiva 2949 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ∀𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
23 iunopn 20528 . . . . . . . . . 10 (((𝐽t 𝑘) ∈ Top ∧ ∀𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘)) → 𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
2414, 22, 23syl2anc 691 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
2512, 24syl5eqelr 2693 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ( 𝑥𝑘) ∈ (𝐽t 𝑘))
2625expr 641 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))
2726ralrimiva 2949 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))
28 elkgen 21149 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → ( 𝑥 ∈ (𝑘Gen‘𝐽) ↔ ( 𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))))
2928adantr 480 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ( 𝑥 ∈ (𝑘Gen‘𝐽) ↔ ( 𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))))
307, 27, 29mpbir2and 959 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → 𝑥 ∈ (𝑘Gen‘𝐽))
3130ex 449 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)))
3231alrimiv 1842 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)))
33 inss1 3795 . . . . . 6 (𝑥𝑦) ⊆ 𝑥
34 elssuni 4403 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
3534ad2antrl 760 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥 (𝑘Gen‘𝐽))
36 ssid 3587 . . . . . . . . . . . 12 𝑋𝑋
3736a1i 11 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝑋)
38 elpwi 4117 . . . . . . . . . . . . . . . 16 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
3938ad2antrl 760 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘𝑋)
40 sseqin2 3779 . . . . . . . . . . . . . . 15 (𝑘𝑋 ↔ (𝑋𝑘) = 𝑘)
4139, 40sylib 207 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑋𝑘) = 𝑘)
4238adantr 480 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp) → 𝑘𝑋)
43 resttopon 20775 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
4442, 43sylan2 490 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
45 toponmax 20543 . . . . . . . . . . . . . . 15 ((𝐽t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 ∈ (𝐽t 𝑘))
4644, 45syl 17 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘 ∈ (𝐽t 𝑘))
4741, 46eqeltrd 2688 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑋𝑘) ∈ (𝐽t 𝑘))
4847expr 641 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))
4948ralrimiva 2949 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))
50 elkgen 21149 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → (𝑋 ∈ (𝑘Gen‘𝐽) ↔ (𝑋𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))))
5137, 49, 50mpbir2and 959 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ (𝑘Gen‘𝐽))
52 elssuni 4403 . . . . . . . . . 10 (𝑋 ∈ (𝑘Gen‘𝐽) → 𝑋 (𝑘Gen‘𝐽))
5351, 52syl 17 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 (𝑘Gen‘𝐽))
5453, 6eqssd 3585 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = (𝑘Gen‘𝐽))
5554adantr 480 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑋 = (𝑘Gen‘𝐽))
5635, 55sseqtr4d 3605 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥𝑋)
5733, 56syl5ss 3579 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥𝑦) ⊆ 𝑋)
58 inindir 3793 . . . . . . . 8 ((𝑥𝑦) ∩ 𝑘) = ((𝑥𝑘) ∩ (𝑦𝑘))
5913ad2antll 761 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
60 simplrl 796 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐽))
61 simprr 792 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Comp)
62 kgeni 21150 . . . . . . . . . 10 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑥𝑘) ∈ (𝐽t 𝑘))
6360, 61, 62syl2anc 691 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ (𝐽t 𝑘))
64 simplrr 797 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑦 ∈ (𝑘Gen‘𝐽))
6564, 61, 19syl2anc 691 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑦𝑘) ∈ (𝐽t 𝑘))
66 inopn 20529 . . . . . . . . 9 (((𝐽t 𝑘) ∈ Top ∧ (𝑥𝑘) ∈ (𝐽t 𝑘) ∧ (𝑦𝑘) ∈ (𝐽t 𝑘)) → ((𝑥𝑘) ∩ (𝑦𝑘)) ∈ (𝐽t 𝑘))
6759, 63, 65, 66syl3anc 1318 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑥𝑘) ∩ (𝑦𝑘)) ∈ (𝐽t 𝑘))
6858, 67syl5eqel 2692 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘))
6968expr 641 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))
7069ralrimiva 2949 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))
71 elkgen 21149 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → ((𝑥𝑦) ∈ (𝑘Gen‘𝐽) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
7271adantr 480 . . . . 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)))))
7357, 70, 72mpbir2and 959 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥𝑦) ∈ (𝑘Gen‘𝐽))
7473ralrimivva 2954 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥𝑦) ∈ (𝑘Gen‘𝐽))
75 fvex 6113 . . . 4 (𝑘Gen‘𝐽) ∈ V
76 istopg 20525 . . . 4 ((𝑘Gen‘𝐽) ∈ V → ((𝑘Gen‘𝐽) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥𝑦) ∈ (𝑘Gen‘𝐽))))
7775, 76ax-mp 5 . . 3 ((𝑘Gen‘𝐽) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥𝑦) ∈ (𝑘Gen‘𝐽)))
7832, 74, 77sylanbrc 695 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ Top)
79 istopon 20540 . 2 ((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ↔ ((𝑘Gen‘𝐽) ∈ Top ∧ 𝑋 = (𝑘Gen‘𝐽)))
8078, 54, 79sylanbrc 695 1 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∪ ciun 4455  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  TopOnctopon 20518  Compccmp 20999  𝑘Genckgen 21146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-kgen 21147 This theorem is referenced by:  kgenuni  21152  kgenftop  21153  kgenhaus  21157  kgenidm  21160  kgencn  21169  kgencn3  21171  kgen2cn  21172
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