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Theorem 1stckgen 23549
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 23438 . 2 (𝐽 ∈ 1stω → 𝐽 ∈ Top)
2 difss 4131 . . . . . . . . . 10 ( 𝐽𝑥) ⊆ 𝐽
3 eqid 2726 . . . . . . . . . . 11 𝐽 = 𝐽
431stcelcls 23456 . . . . . . . . . 10 ((𝐽 ∈ 1stω ∧ ( 𝐽𝑥) ⊆ 𝐽) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
52, 4mpan2 689 . . . . . . . . 9 (𝐽 ∈ 1stω → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
65adantr 479 . . . . . . . 8 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
71adantr 479 . . . . . . . . . . . . . 14 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 ∈ Top)
87adantr 479 . . . . . . . . . . . . 13 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝐽 ∈ Top)
9 toptopon2 22911 . . . . . . . . . . . . 13 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
108, 9sylib 217 . . . . . . . . . . . 12 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝐽 ∈ (TopOn‘ 𝐽))
11 simprr 771 . . . . . . . . . . . 12 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓(⇝𝑡𝐽)𝑦)
12 lmcl 23292 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 𝐽)
1310, 11, 12syl2anc 582 . . . . . . . . . . 11 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 𝐽)
14 nnuz 12917 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
15 vex 3466 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
1615rnex 7923 . . . . . . . . . . . . . . . 16 ran 𝑓 ∈ V
17 vsnex 5435 . . . . . . . . . . . . . . . 16 {𝑦} ∈ V
1816, 17unex 7754 . . . . . . . . . . . . . . 15 (ran 𝑓 ∪ {𝑦}) ∈ V
19 resttop 23155 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ∈ V) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top)
208, 18, 19sylancl 584 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top)
21 toptopon2 22911 . . . . . . . . . . . . . 14 ((𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top ↔ (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘ (𝐽t (ran 𝑓 ∪ {𝑦}))))
2220, 21sylib 217 . . . . . . . . . . . . 13 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘ (𝐽t (ran 𝑓 ∪ {𝑦}))))
23 1zzd 12645 . . . . . . . . . . . . 13 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 1 ∈ ℤ)
24 eqid 2726 . . . . . . . . . . . . . . 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 4174 . . . . . . . . . . . . . . . . 17 {𝑦} ⊆ (ran 𝑓 ∪ {𝑦})
27 vex 3466 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
2827snss 4794 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ran 𝑓 ∪ {𝑦}) ↔ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦}))
2926, 28mpbir 230 . . . . . . . . . . . . . . . 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 6728 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶( 𝐽𝑥) → 𝑓 Fn ℕ)
3231ad2antrl 726 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓 Fn ℕ)
33 dffn3 6740 . . . . . . . . . . . . . . . . 17 (𝑓 Fn ℕ ↔ 𝑓:ℕ⟶ran 𝑓)
3432, 33sylib 217 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶ran 𝑓)
35 ssun1 4173 . . . . . . . . . . . . . . . 16 ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})
36 fss 6744 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶ran 𝑓 ∧ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
3734, 35, 36sylancl 584 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
3824, 14, 25, 8, 30, 23, 37lmss 23293 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝑓(⇝𝑡𝐽)𝑦𝑓(⇝𝑡‘(𝐽t (ran 𝑓 ∪ {𝑦})))𝑦))
3911, 38mpbid 231 . . . . . . . . . . . . 13 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓(⇝𝑡‘(𝐽t (ran 𝑓 ∪ {𝑦})))𝑦)
4037ffvelcdmda 7098 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ (ran 𝑓 ∪ {𝑦}))
41 simprl 769 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶( 𝐽𝑥))
4241ffvelcdmda 7098 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ ( 𝐽𝑥))
4342eldifbd 3960 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → ¬ (𝑓𝑘) ∈ 𝑥)
4440, 43eldifd 3958 . . . . . . . . . . . . 13 ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
45 difin 4263 . . . . . . . . . . . . . . 15 ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥)
46 frn 6735 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℕ⟶( 𝐽𝑥) → ran 𝑓 ⊆ ( 𝐽𝑥))
4746ad2antrl 726 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ran 𝑓 ⊆ ( 𝐽𝑥))
4847difss2d 4134 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ran 𝑓 𝐽)
4913snssd 4818 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → {𝑦} ⊆ 𝐽)
5048, 49unssd 4187 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ⊆ 𝐽)
513restuni 23157 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ⊆ 𝐽) → (ran 𝑓 ∪ {𝑦}) = (𝐽t (ran 𝑓 ∪ {𝑦})))
528, 50, 51syl2anc 582 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) = (𝐽t (ran 𝑓 ∪ {𝑦})))
5352difeq1d 4120 . . . . . . . . . . . . . . 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, 53eqtr3id 2780 . . . . . . . . . . . . . 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 4202 . . . . . . . . . . . . . . . 16 ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) = (𝑥 ∩ (ran 𝑓 ∪ {𝑦}))
56 simplr 767 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑥 ∈ (𝑘Gen‘𝐽))
57 fss 6744 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ⟶( 𝐽𝑥) ∧ ( 𝐽𝑥) ⊆ 𝐽) → 𝑓:ℕ⟶ 𝐽)
5841, 2, 57sylancl 584 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶ 𝐽)
5910, 58, 111stckgenlem 23548 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Comp)
60 kgeni 23532 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Comp) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6156, 59, 60syl2anc 582 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6255, 61eqeltrid 2830 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
63 eqid 2726 . . . . . . . . . . . . . . . 16 (𝐽t (ran 𝑓 ∪ {𝑦})) = (𝐽t (ran 𝑓 ∪ {𝑦}))
6463opncld 23028 . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . 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 23298 . . . . . . . . . . . 12 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
6867eldifbd 3960 . . . . . . . . . . 11 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ¬ 𝑦𝑥)
6913, 68eldifd 3958 . . . . . . . . . 10 (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ ( 𝐽𝑥))
7069ex 411 . . . . . . . . 9 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 ∈ ( 𝐽𝑥)))
7170exlimdv 1929 . . . . . . . 8 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 ∈ ( 𝐽𝑥)))
726, 71sylbid 239 . . . . . . 7 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) → 𝑦 ∈ ( 𝐽𝑥)))
7372ssrdv 3985 . . . . . 6 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥))
743iscld4 23060 . . . . . . 7 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ⊆ 𝐽) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥)))
757, 2, 74sylancl 584 . . . . . 6 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥)))
7673, 75mpbird 256 . . . . 5 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
77 elssuni 4945 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
7877adantl 480 . . . . . . 7 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 (𝑘Gen‘𝐽))
793kgenuni 23534 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 = (𝑘Gen‘𝐽))
807, 79syl 17 . . . . . . 7 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 = (𝑘Gen‘𝐽))
8178, 80sseqtrrd 4021 . . . . . 6 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 𝐽)
823isopn2 23027 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → (𝑥𝐽 ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
837, 81, 82syl2anc 582 . . . . 5 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥𝐽 ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
8476, 83mpbird 256 . . . 4 ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥𝐽)
8584ex 411 . . 3 (𝐽 ∈ 1stω → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥𝐽))
8685ssrdv 3985 . 2 (𝐽 ∈ 1stω → (𝑘Gen‘𝐽) ⊆ 𝐽)
87 iskgen2 23543 . 2 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
881, 86, 87sylanbrc 581 1 (𝐽 ∈ 1stω → 𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wex 1774  wcel 2099  Vcvv 3462  cdif 3944  cun 3945  cin 3946  wss 3947  {csn 4633   cuni 4913   class class class wbr 5153  ran crn 5683   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424  1c1 11159  cn 12264  t crest 17435  Topctop 22886  TopOnctopon 22903  Clsdccld 23011  clsccl 23013  𝑡clm 23221  Compccmp 23381  1stωc1stc 23432  𝑘Genckgen 23528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cc 10478  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-pm 8858  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fi 9454  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-n0 12525  df-z 12611  df-uz 12875  df-fz 13539  df-rest 17437  df-topgen 17458  df-top 22887  df-topon 22904  df-bases 22940  df-cld 23014  df-ntr 23015  df-cls 23016  df-lm 23224  df-cmp 23382  df-1stc 23434  df-kgen 23529
This theorem is referenced by: (None)
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