MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kgentopon Structured version   Visualization version   GIF version

Theorem kgentopon 23033
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 4915 . . . . . . 7 (𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
2 kgenval 23030 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) = {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))})
3 ssrab2 4076 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝑋 ∣ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑥𝑘) ∈ (𝐽t 𝑘))} ⊆ 𝒫 𝑋
42, 3eqsstrdi 4035 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝒫 𝑋)
5 sspwuni 5102 . . . . . . . 8 ((𝑘Gen‘𝐽) ⊆ 𝒫 𝑋 (𝑘Gen‘𝐽) ⊆ 𝑋)
64, 5sylib 217 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ⊆ 𝑋)
71, 6sylan9ssr 3995 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → 𝑥𝑋)
8 iunin2 5073 . . . . . . . . . 10 𝑦𝑥 (𝑘𝑦) = (𝑘 𝑦𝑥 𝑦)
9 uniiun 5060 . . . . . . . . . . 11 𝑥 = 𝑦𝑥 𝑦
109ineq2i 4208 . . . . . . . . . 10 (𝑘 𝑥) = (𝑘 𝑦𝑥 𝑦)
11 incom 4200 . . . . . . . . . 10 (𝑘 𝑥) = ( 𝑥𝑘)
128, 10, 113eqtr2i 2766 . . . . . . . . 9 𝑦𝑥 (𝑘𝑦) = ( 𝑥𝑘)
13 cmptop 22890 . . . . . . . . . . 11 ((𝐽t 𝑘) ∈ Comp → (𝐽t 𝑘) ∈ Top)
1413ad2antll 727 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
15 incom 4200 . . . . . . . . . . . 12 (𝑦𝑘) = (𝑘𝑦)
16 simplr 767 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ⊆ (𝑘Gen‘𝐽))
1716sselda 3981 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → 𝑦 ∈ (𝑘Gen‘𝐽))
18 simplrr 776 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝐽t 𝑘) ∈ Comp)
19 kgeni 23032 . . . . . . . . . . . . 13 ((𝑦 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑦𝑘) ∈ (𝐽t 𝑘))
2017, 18, 19syl2anc 584 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝑦𝑘) ∈ (𝐽t 𝑘))
2115, 20eqeltrrid 2838 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) ∧ 𝑦𝑥) → (𝑘𝑦) ∈ (𝐽t 𝑘))
2221ralrimiva 3146 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ∀𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
23 iunopn 22391 . . . . . . . . . 10 (((𝐽t 𝑘) ∈ Top ∧ ∀𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘)) → 𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
2414, 22, 23syl2anc 584 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑦𝑥 (𝑘𝑦) ∈ (𝐽t 𝑘))
2512, 24eqeltrrid 2838 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ( 𝑥𝑘) ∈ (𝐽t 𝑘))
2625expr 457 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))
2726ralrimiva 3146 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ( 𝑥𝑘) ∈ (𝐽t 𝑘)))
28 elkgen 23031 . . . . . . 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 711 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (𝑘Gen‘𝐽)) → 𝑥 ∈ (𝑘Gen‘𝐽))
3130ex 413 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)))
3231alrimiv 1930 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥(𝑥 ⊆ (𝑘Gen‘𝐽) → 𝑥 ∈ (𝑘Gen‘𝐽)))
33 inss1 4227 . . . . . 6 (𝑥𝑦) ⊆ 𝑥
34 elssuni 4940 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
3534ad2antrl 726 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥 (𝑘Gen‘𝐽))
36 ssidd 4004 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝑋)
37 elpwi 4608 . . . . . . . . . . . . . . . 16 (𝑘 ∈ 𝒫 𝑋𝑘𝑋)
3837ad2antrl 726 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑘𝑋)
39 sseqin2 4214 . . . . . . . . . . . . . . 15 (𝑘𝑋 ↔ (𝑋𝑘) = 𝑘)
4038, 39sylib 217 . . . . . . . . . . . . . 14 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑋𝑘) = 𝑘)
4137adantr 481 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp) → 𝑘𝑋)
42 resttopon 22656 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘𝑋) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
4341, 42sylan2 593 . . . . . . . . . . . . . . 15 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ (TopOn‘𝑘))
44 toponmax 22419 . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . 13 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑋𝑘) ∈ (𝐽t 𝑘))
4746expr 457 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))
4847ralrimiva 3146 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))
49 elkgen 23031 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → (𝑋 ∈ (𝑘Gen‘𝐽) ↔ (𝑋𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → (𝑋𝑘) ∈ (𝐽t 𝑘)))))
5036, 48, 49mpbir2and 711 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ (𝑘Gen‘𝐽))
51 elssuni 4940 . . . . . . . . . 10 (𝑋 ∈ (𝑘Gen‘𝐽) → 𝑋 (𝑘Gen‘𝐽))
5250, 51syl 17 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 (𝑘Gen‘𝐽))
5352, 6eqssd 3998 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = (𝑘Gen‘𝐽))
5453adantr 481 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑋 = (𝑘Gen‘𝐽))
5535, 54sseqtrrd 4022 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → 𝑥𝑋)
5633, 55sstrid 3992 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥𝑦) ⊆ 𝑋)
57 inindir 4226 . . . . . . . 8 ((𝑥𝑦) ∩ 𝑘) = ((𝑥𝑘) ∩ (𝑦𝑘))
5813ad2antll 727 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Top)
59 simplrl 775 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑘Gen‘𝐽))
60 simprr 771 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝐽t 𝑘) ∈ Comp)
61 kgeni 23032 . . . . . . . . . 10 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝑘) ∈ Comp) → (𝑥𝑘) ∈ (𝐽t 𝑘))
6259, 60, 61syl2anc 584 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑥𝑘) ∈ (𝐽t 𝑘))
63 simplrr 776 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → 𝑦 ∈ (𝑘Gen‘𝐽))
6463, 60, 19syl2anc 584 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → (𝑦𝑘) ∈ (𝐽t 𝑘))
65 inopn 22392 . . . . . . . . 9 (((𝐽t 𝑘) ∈ Top ∧ (𝑥𝑘) ∈ (𝐽t 𝑘) ∧ (𝑦𝑘) ∈ (𝐽t 𝑘)) → ((𝑥𝑘) ∩ (𝑦𝑘)) ∈ (𝐽t 𝑘))
6658, 62, 64, 65syl3anc 1371 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑥𝑘) ∩ (𝑦𝑘)) ∈ (𝐽t 𝑘))
6757, 66eqeltrid 2837 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ (𝑘 ∈ 𝒫 𝑋 ∧ (𝐽t 𝑘) ∈ Comp)) → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘))
6867expr 457 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))
6968ralrimiva 3146 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))
70 elkgen 23031 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → ((𝑥𝑦) ∈ (𝑘Gen‘𝐽) ↔ ((𝑥𝑦) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽t 𝑘) ∈ Comp → ((𝑥𝑦) ∩ 𝑘) ∈ (𝐽t 𝑘)))))
7170adantr 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)))))
7256, 69, 71mpbir2and 711 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ∈ (𝑘Gen‘𝐽) ∧ 𝑦 ∈ (𝑘Gen‘𝐽))) → (𝑥𝑦) ∈ (𝑘Gen‘𝐽))
7372ralrimivva 3200 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∀𝑥 ∈ (𝑘Gen‘𝐽)∀𝑦 ∈ (𝑘Gen‘𝐽)(𝑥𝑦) ∈ (𝑘Gen‘𝐽))
74 fvex 6901 . . . 4 (𝑘Gen‘𝐽) ∈ V
75 istopg 22388 . . . 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 583 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ Top)
78 istopon 22405 . 2 ((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ↔ ((𝑘Gen‘𝐽) ∈ Top ∧ 𝑋 = (𝑘Gen‘𝐽)))
7977, 53, 78sylanbrc 583 1 (𝐽 ∈ (TopOn‘𝑋) → (𝑘Gen‘𝐽) ∈ (TopOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  wral 3061  {crab 3432  Vcvv 3474  cin 3946  wss 3947  𝒫 cpw 4601   cuni 4907   ciun 4996  cfv 6540  (class class class)co 7405  t crest 17362  Topctop 22386  TopOnctopon 22403  Compccmp 22881  𝑘Genckgen 23028
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-en 8936  df-fin 8939  df-fi 9402  df-rest 17364  df-topgen 17385  df-top 22387  df-topon 22404  df-bases 22440  df-cmp 22882  df-kgen 23029
This theorem is referenced by:  kgenuni  23034  kgenftop  23035  kgenhaus  23039  kgenidm  23042  kgencn  23051  kgencn3  23053  kgen2cn  23054
  Copyright terms: Public domain W3C validator