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Theorem 1stckgen 21846
Description: A first-countable space is compactly generated. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
1stckgen (𝐽 ∈ 1st𝜔 → 𝐽 ∈ ran 𝑘Gen)

Proof of Theorem 1stckgen
Dummy variables 𝑘 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stctop 21735 . 2 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
2 difss 4029 . . . . . . . . . 10 ( 𝐽𝑥) ⊆ 𝐽
3 eqid 2795 . . . . . . . . . . 11 𝐽 = 𝐽
431stcelcls 21753 . . . . . . . . . 10 ((𝐽 ∈ 1st𝜔 ∧ ( 𝐽𝑥) ⊆ 𝐽) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
52, 4mpan2 687 . . . . . . . . 9 (𝐽 ∈ 1st𝜔 → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
65adantr 481 . . . . . . . 8 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
71adantr 481 . . . . . . . . . . . . . 14 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 ∈ Top)
87adantr 481 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝐽 ∈ Top)
9 toptopon2 21210 . . . . . . . . . . . . 13 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
108, 9sylib 219 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝐽 ∈ (TopOn‘ 𝐽))
11 simprr 769 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓(⇝𝑡𝐽)𝑦)
12 lmcl 21589 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 𝐽)
1310, 11, 12syl2anc 584 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 𝐽)
14 nnuz 12130 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
15 vex 3440 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
1615rnex 7473 . . . . . . . . . . . . . . . 16 ran 𝑓 ∈ V
17 snex 5223 . . . . . . . . . . . . . . . 16 {𝑦} ∈ V
1816, 17unex 7326 . . . . . . . . . . . . . . 15 (ran 𝑓 ∪ {𝑦}) ∈ V
19 resttop 21452 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ∈ V) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top)
208, 18, 19sylancl 586 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top)
21 toptopon2 21210 . . . . . . . . . . . . . 14 ((𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top ↔ (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘ (𝐽t (ran 𝑓 ∪ {𝑦}))))
2220, 21sylib 219 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘ (𝐽t (ran 𝑓 ∪ {𝑦}))))
23 1zzd 11862 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 1 ∈ ℤ)
24 eqid 2795 . . . . . . . . . . . . . . 15 (𝐽t (ran 𝑓 ∪ {𝑦})) = (𝐽t (ran 𝑓 ∪ {𝑦}))
2518a1i 11 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ∈ V)
26 ssun2 4070 . . . . . . . . . . . . . . . . 17 {𝑦} ⊆ (ran 𝑓 ∪ {𝑦})
27 vex 3440 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
2827snss 4625 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ran 𝑓 ∪ {𝑦}) ↔ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦}))
2926, 28mpbir 232 . . . . . . . . . . . . . . . 16 𝑦 ∈ (ran 𝑓 ∪ {𝑦})
3029a1i 11 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ (ran 𝑓 ∪ {𝑦}))
31 ffn 6382 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶( 𝐽𝑥) → 𝑓 Fn ℕ)
3231ad2antrl 724 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓 Fn ℕ)
33 dffn3 6393 . . . . . . . . . . . . . . . . 17 (𝑓 Fn ℕ ↔ 𝑓:ℕ⟶ran 𝑓)
3432, 33sylib 219 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶ran 𝑓)
35 ssun1 4069 . . . . . . . . . . . . . . . 16 ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})
36 fss 6395 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶ran 𝑓 ∧ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
3734, 35, 36sylancl 586 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
3824, 14, 25, 8, 30, 23, 37lmss 21590 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝑓(⇝𝑡𝐽)𝑦𝑓(⇝𝑡‘(𝐽t (ran 𝑓 ∪ {𝑦})))𝑦))
3911, 38mpbid 233 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓(⇝𝑡‘(𝐽t (ran 𝑓 ∪ {𝑦})))𝑦)
4037ffvelrnda 6716 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ (ran 𝑓 ∪ {𝑦}))
41 simprl 767 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶( 𝐽𝑥))
4241ffvelrnda 6716 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ ( 𝐽𝑥))
4342eldifbd 3872 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → ¬ (𝑓𝑘) ∈ 𝑥)
4440, 43eldifd 3870 . . . . . . . . . . . . 13 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
45 difin 4158 . . . . . . . . . . . . . . 15 ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥)
46 frn 6388 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℕ⟶( 𝐽𝑥) → ran 𝑓 ⊆ ( 𝐽𝑥))
4746ad2antrl 724 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ran 𝑓 ⊆ ( 𝐽𝑥))
4847difss2d 4032 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ran 𝑓 𝐽)
4913snssd 4649 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → {𝑦} ⊆ 𝐽)
5048, 49unssd 4083 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ⊆ 𝐽)
513restuni 21454 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ⊆ 𝐽) → (ran 𝑓 ∪ {𝑦}) = (𝐽t (ran 𝑓 ∪ {𝑦})))
528, 50, 51syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) = (𝐽t (ran 𝑓 ∪ {𝑦})))
5352difeq1d 4019 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ( (𝐽t (ran 𝑓 ∪ {𝑦})) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)))
5445, 53syl5eqr 2845 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥) = ( (𝐽t (ran 𝑓 ∪ {𝑦})) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)))
55 incom 4099 . . . . . . . . . . . . . . . 16 ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) = (𝑥 ∩ (ran 𝑓 ∪ {𝑦}))
56 simplr 765 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑥 ∈ (𝑘Gen‘𝐽))
57 fss 6395 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ⟶( 𝐽𝑥) ∧ ( 𝐽𝑥) ⊆ 𝐽) → 𝑓:ℕ⟶ 𝐽)
5841, 2, 57sylancl 586 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶ 𝐽)
5910, 58, 111stckgenlem 21845 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Comp)
60 kgeni 21829 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Comp) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6156, 59, 60syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6255, 61syl5eqel 2887 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
63 eqid 2795 . . . . . . . . . . . . . . . 16 (𝐽t (ran 𝑓 ∪ {𝑦})) = (𝐽t (ran 𝑓 ∪ {𝑦}))
6463opncld 21325 . . . . . . . . . . . . . . 15 (((𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top ∧ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) ∈ (𝐽t (ran 𝑓 ∪ {𝑦}))) → ( (𝐽t (ran 𝑓 ∪ {𝑦})) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) ∈ (Clsd‘(𝐽t (ran 𝑓 ∪ {𝑦}))))
6520, 62, 64syl2anc 584 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ( (𝐽t (ran 𝑓 ∪ {𝑦})) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) ∈ (Clsd‘(𝐽t (ran 𝑓 ∪ {𝑦}))))
6654, 65eqeltrd 2883 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥) ∈ (Clsd‘(𝐽t (ran 𝑓 ∪ {𝑦}))))
6714, 22, 23, 39, 44, 66lmcld 21595 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
6867eldifbd 3872 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ¬ 𝑦𝑥)
6913, 68eldifd 3870 . . . . . . . . . 10 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ ( 𝐽𝑥))
7069ex 413 . . . . . . . . 9 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 ∈ ( 𝐽𝑥)))
7170exlimdv 1911 . . . . . . . 8 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 ∈ ( 𝐽𝑥)))
726, 71sylbid 241 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) → 𝑦 ∈ ( 𝐽𝑥)))
7372ssrdv 3895 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥))
743iscld4 21357 . . . . . . 7 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ⊆ 𝐽) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥)))
757, 2, 74sylancl 586 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥)))
7673, 75mpbird 258 . . . . 5 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
77 elssuni 4774 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
7877adantl 482 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 (𝑘Gen‘𝐽))
793kgenuni 21831 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 = (𝑘Gen‘𝐽))
807, 79syl 17 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 = (𝑘Gen‘𝐽))
8178, 80sseqtr4d 3929 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 𝐽)
823isopn2 21324 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → (𝑥𝐽 ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
837, 81, 82syl2anc 584 . . . . 5 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥𝐽 ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
8476, 83mpbird 258 . . . 4 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥𝐽)
8584ex 413 . . 3 (𝐽 ∈ 1st𝜔 → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥𝐽))
8685ssrdv 3895 . 2 (𝐽 ∈ 1st𝜔 → (𝑘Gen‘𝐽) ⊆ 𝐽)
87 iskgen2 21840 . 2 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
881, 86, 87sylanbrc 583 1 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wex 1761  wcel 2081  Vcvv 3437  cdif 3856  cun 3857  cin 3858  wss 3859  {csn 4472   cuni 4745   class class class wbr 4962  ran crn 5444   Fn wfn 6220  wf 6221  cfv 6225  (class class class)co 7016  1c1 10384  cn 11486  t crest 16523  Topctop 21185  TopOnctopon 21202  Clsdccld 21308  clsccl 21310  𝑡clm 21518  Compccmp 21678  1st𝜔c1stc 21729  𝑘Genckgen 21825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cc 9703  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fi 8721  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-n0 11746  df-z 11830  df-uz 12094  df-fz 12743  df-rest 16525  df-topgen 16546  df-top 21186  df-topon 21203  df-bases 21238  df-cld 21311  df-ntr 21312  df-cls 21313  df-lm 21521  df-cmp 21679  df-1stc 21731  df-kgen 21826
This theorem is referenced by: (None)
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