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Theorem kgentopon 21260
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 4429 . . . . . . 7 (𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
2 kgenval 21257 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})
3 ssrab2 3671 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))} ⊆ 𝒫 𝑋
42, 3syl6eqss 3639 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝒫 𝑋)
5 sspwuni 4582 . . . . . . . 8 ((𝑘Gen‘𝐽) ⊆ 𝒫 𝑋 (𝑘Gen‘𝐽) ⊆ 𝑋)
64, 5sylib 208 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝑋)
71, 6sylan9ssr 3601 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → 𝑥𝑋)
8 iunin2 4555 . . . . . . . . . 10 𝑦𝑥 (𝑘𝑦) = (𝑘 𝑦𝑥 𝑦)
9 uniiun 4544 . . . . . . . . . . 11 𝑥 = 𝑦𝑥 𝑦
109ineq2i 3794 . . . . . . . . . 10 (𝑘 𝑥) = (𝑘 𝑦𝑥 𝑦)
11 incom 3788 . . . . . . . . . 10 (𝑘 𝑥) = ( 𝑥𝑘)
128, 10, 113eqtr2i 2649 . . . . . . . . 9 𝑦𝑥 (𝑘𝑦) = ( 𝑥𝑘)
13 cmptop 21117 . . . . . . . . . . 11 ((𝐽t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Top)
1413ad2antll 764 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
15 incom 3788 . . . . . . . . . . . 12 (𝑦𝑘) = (𝑘𝑦)
16 simplr 791 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ⊆ (𝑘Gen‘𝐽))
1716sselda 3587 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → 𝑦 ∈ (𝑘Gen‘𝐽))
18 simplrr 800 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝐽t 𝑘) ∈ Comp)
19 kgeni 21259 . . . . . . . . . . . . 13 ((𝑦 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑦𝑘) ∈ (𝐽t 𝑘))
2017, 18, 19syl2anc 692 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝑦𝑘) ∈ (𝐽t 𝑘))
2115, 20syl5eqelr 2703 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝑘𝑦) ∈ (𝐽t 𝑘))
2221ralrimiva 2961 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ∀𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
23 iunopn 20631 . . . . . . . . . 10 (((𝐽t 𝑘) ∈ Top ∧ ∀𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘)) → 𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
2414, 22, 23syl2anc 692 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
2512, 24syl5eqelr 2703 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ( 𝑥𝑘) ∈ (𝐽t 𝑘))
2625expr 642 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))
2726ralrimiva 2961 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))
28 elkgen 21258 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → ( 𝑥 ∈ (𝑘Gen‘𝐽) ↔ ( 𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))))
2928adantr 481 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ( 𝑥 ∈ (𝑘Gen‘𝐽) ↔ ( 𝑥𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))))
307, 27, 29mpbir2and 956 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → 𝑥 ∈ (𝑘Gen‘𝐽))
3130ex 450 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)))
3231alrimiv 1852 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)))
33 inss1 3816 . . . . . 6 (𝑥𝑦) ⊆ 𝑥
34 elssuni 4438 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
3534ad2antrl 763 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥 (𝑘Gen‘𝐽))
36 ssid 3608 . . . . . . . . . . . 12 𝑋𝑋
3736a1i 11 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝑋)
38 elpwi 4145 . . . . . . . . . . . . . . . 16 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
3938ad2antrl 763 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘𝑋)
40 sseqin2 3800 . . . . . . . . . . . . . . 15 (𝑘𝑋 ↔ (𝑋𝑘) = 𝑘)
4139, 40sylib 208 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑋𝑘) = 𝑘)
4238adantr 481 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp) → 𝑘𝑋)
43 resttopon 20884 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
4442, 43sylan2 491 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
45 toponmax 20648 . . . . . . . . . . . . . . 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 2698 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑋𝑘) ∈ (𝐽t 𝑘))
4847expr 642 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))
4948ralrimiva 2961 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))
50 elkgen 21258 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → (𝑋 ∈ (𝑘Gen‘𝐽) ↔ (𝑋𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))))
5137, 49, 50mpbir2and 956 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ (𝑘Gen‘𝐽))
52 elssuni 4438 . . . . . . . . . 10 (𝑋 ∈ (𝑘Gen‘𝐽) → 𝑋 (𝑘Gen‘𝐽))
5351, 52syl 17 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 (𝑘Gen‘𝐽))
5453, 6eqssd 3604 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = (𝑘Gen‘𝐽))
5554adantr 481 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑋 = (𝑘Gen‘𝐽))
5635, 55sseqtr4d 3626 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥𝑋)
5733, 56syl5ss 3598 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥𝑦) ⊆ 𝑋)
58 inindir 3814 . . . . . . . 8 ((𝑥𝑦) ∩ 𝑘) = ((𝑥𝑘) ∩ (𝑦𝑘))
5913ad2antll 764 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
60 simplrl 799 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐽))
61 simprr 795 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Comp)
62 kgeni 21259 . . . . . . . . . 10 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑥𝑘) ∈ (𝐽t 𝑘))
6360, 61, 62syl2anc 692 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ (𝐽t 𝑘))
64 simplrr 800 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑦 ∈ (𝑘Gen‘𝐽))
6564, 61, 19syl2anc 692 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑦𝑘) ∈ (𝐽t 𝑘))
66 inopn 20632 . . . . . . . . 9 (((𝐽t 𝑘) ∈ Top ∧ (𝑥𝑘) ∈ (𝐽t 𝑘) ∧ (𝑦𝑘) ∈ (𝐽t 𝑘)) → ((𝑥𝑘) ∩ (𝑦𝑘)) ∈ (𝐽t 𝑘))
6759, 63, 65, 66syl3anc 1323 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑥𝑘) ∩ (𝑦𝑘)) ∈ (𝐽t 𝑘))
6858, 67syl5eqel 2702 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘))
6968expr 642 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))
7069ralrimiva 2961 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))
71 elkgen 21258 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → ((𝑥𝑦) ∈ (𝑘Gen‘𝐽) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
7271adantr 481 . . . . 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 956 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥𝑦) ∈ (𝑘Gen‘𝐽))
7473ralrimivva 2966 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥𝑦) ∈ (𝑘Gen‘𝐽))
75 fvex 6163 . . . 4 (𝑘Gen‘𝐽) ∈ V
76 istopg 20628 . . . 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 697 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ Top)
79 istopon 20645 . 2 ((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ↔ ((𝑘Gen‘𝐽) ∈ Top ∧ 𝑋 = (𝑘Gen‘𝐽)))
8078, 54, 79sylanbrc 697 1 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  cin 3558  wss 3559  𝒫 cpw 4135   cuni 4407   ciun 4490  cfv 5852  (class class class)co 6610  t crest 16009  Topctop 20626  TopOnctopon 20643  Compccmp 21108  𝑘Genckgen 21255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-oadd 7516  df-er 7694  df-en 7907  df-fin 7910  df-fi 8268  df-rest 16011  df-topgen 16032  df-top 20627  df-topon 20644  df-bases 20670  df-cmp 21109  df-kgen 21256
This theorem is referenced by:  kgenuni  21261  kgenftop  21262  kgenhaus  21266  kgenidm  21269  kgencn  21278  kgencn3  21280  kgen2cn  21281
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