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Theorem 1stckgen 21530
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 21419 . 2 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
2 difss 3868 . . . . . . . . . 10 ( 𝐽𝑥) ⊆ 𝐽
3 eqid 2748 . . . . . . . . . . 11 𝐽 = 𝐽
431stcelcls 21437 . . . . . . . . . 10 ((𝐽 ∈ 1st𝜔 ∧ ( 𝐽𝑥) ⊆ 𝐽) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
52, 4mpan2 709 . . . . . . . . 9 (𝐽 ∈ 1st𝜔 → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
65adantr 472 . . . . . . . 8 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
71adantr 472 . . . . . . . . . . . . . 14 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 ∈ Top)
87adantr 472 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝐽 ∈ Top)
93toptopon 20895 . . . . . . . . . . . . 13 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
108, 9sylib 208 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝐽 ∈ (TopOn‘ 𝐽))
11 simprr 813 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓(⇝𝑡𝐽)𝑦)
12 lmcl 21274 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 𝐽)
1310, 11, 12syl2anc 696 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 𝐽)
14 nnuz 11887 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
15 vex 3331 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
1615rnex 7253 . . . . . . . . . . . . . . . 16 ran 𝑓 ∈ V
17 snex 5045 . . . . . . . . . . . . . . . 16 {𝑦} ∈ V
1816, 17unex 7109 . . . . . . . . . . . . . . 15 (ran 𝑓 ∪ {𝑦}) ∈ V
19 resttop 21137 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ∈ V) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top)
208, 18, 19sylancl 697 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top)
21 eqid 2748 . . . . . . . . . . . . . . 15 (𝐽t (ran 𝑓 ∪ {𝑦})) = (𝐽t (ran 𝑓 ∪ {𝑦}))
2221toptopon 20895 . . . . . . . . . . . . . 14 ((𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top ↔ (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘ (𝐽t (ran 𝑓 ∪ {𝑦}))))
2320, 22sylib 208 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘ (𝐽t (ran 𝑓 ∪ {𝑦}))))
24 1zzd 11571 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 1 ∈ ℤ)
25 eqid 2748 . . . . . . . . . . . . . . 15 (𝐽t (ran 𝑓 ∪ {𝑦})) = (𝐽t (ran 𝑓 ∪ {𝑦}))
2618a1i 11 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ∈ V)
27 ssun2 3908 . . . . . . . . . . . . . . . . 17 {𝑦} ⊆ (ran 𝑓 ∪ {𝑦})
28 vex 3331 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
2928snss 4448 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ran 𝑓 ∪ {𝑦}) ↔ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦}))
3027, 29mpbir 221 . . . . . . . . . . . . . . . 16 𝑦 ∈ (ran 𝑓 ∪ {𝑦})
3130a1i 11 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ (ran 𝑓 ∪ {𝑦}))
32 ffn 6194 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶( 𝐽𝑥) → 𝑓 Fn ℕ)
3332ad2antrl 766 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓 Fn ℕ)
34 dffn3 6203 . . . . . . . . . . . . . . . . 17 (𝑓 Fn ℕ ↔ 𝑓:ℕ⟶ran 𝑓)
3533, 34sylib 208 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶ran 𝑓)
36 ssun1 3907 . . . . . . . . . . . . . . . 16 ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})
37 fss 6205 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶ran 𝑓 ∧ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
3835, 36, 37sylancl 697 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
3925, 14, 26, 8, 31, 24, 38lmss 21275 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝑓(⇝𝑡𝐽)𝑦𝑓(⇝𝑡‘(𝐽t (ran 𝑓 ∪ {𝑦})))𝑦))
4011, 39mpbid 222 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓(⇝𝑡‘(𝐽t (ran 𝑓 ∪ {𝑦})))𝑦)
4138ffvelrnda 6510 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ (ran 𝑓 ∪ {𝑦}))
42 simprl 811 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶( 𝐽𝑥))
4342ffvelrnda 6510 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ ( 𝐽𝑥))
4443eldifbd 3716 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → ¬ (𝑓𝑘) ∈ 𝑥)
4541, 44eldifd 3714 . . . . . . . . . . . . 13 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
46 difin 3992 . . . . . . . . . . . . . . 15 ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥)
47 frn 6202 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℕ⟶( 𝐽𝑥) → ran 𝑓 ⊆ ( 𝐽𝑥))
4847ad2antrl 766 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ran 𝑓 ⊆ ( 𝐽𝑥))
4948difss2d 3871 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ran 𝑓 𝐽)
5013snssd 4473 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → {𝑦} ⊆ 𝐽)
5149, 50unssd 3920 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ⊆ 𝐽)
523restuni 21139 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ⊆ 𝐽) → (ran 𝑓 ∪ {𝑦}) = (𝐽t (ran 𝑓 ∪ {𝑦})))
538, 51, 52syl2anc 696 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) = (𝐽t (ran 𝑓 ∪ {𝑦})))
5453difeq1d 3858 . . . . . . . . . . . . . . 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)))
5546, 54syl5eqr 2796 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥) = ( (𝐽t (ran 𝑓 ∪ {𝑦})) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)))
56 incom 3936 . . . . . . . . . . . . . . . 16 ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) = (𝑥 ∩ (ran 𝑓 ∪ {𝑦}))
57 simplr 809 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑥 ∈ (𝑘Gen‘𝐽))
58 fss 6205 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ⟶( 𝐽𝑥) ∧ ( 𝐽𝑥) ⊆ 𝐽) → 𝑓:ℕ⟶ 𝐽)
5942, 2, 58sylancl 697 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶ 𝐽)
6010, 59, 111stckgenlem 21529 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Comp)
61 kgeni 21513 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Comp) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6257, 60, 61syl2anc 696 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6356, 62syl5eqel 2831 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6421opncld 21010 . . . . . . . . . . . . . . 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, 63, 64syl2anc 696 . . . . . . . . . . . . . 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}))))
6655, 65eqeltrd 2827 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥) ∈ (Clsd‘(𝐽t (ran 𝑓 ∪ {𝑦}))))
6714, 23, 24, 40, 45, 66lmcld 21280 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
6867eldifbd 3716 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ¬ 𝑦𝑥)
6913, 68eldifd 3714 . . . . . . . . . 10 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ ( 𝐽𝑥))
7069ex 449 . . . . . . . . 9 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 ∈ ( 𝐽𝑥)))
7170exlimdv 1998 . . . . . . . 8 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 ∈ ( 𝐽𝑥)))
726, 71sylbid 230 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) → 𝑦 ∈ ( 𝐽𝑥)))
7372ssrdv 3738 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥))
743iscld4 21042 . . . . . . 7 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ⊆ 𝐽) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥)))
757, 2, 74sylancl 697 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥)))
7673, 75mpbird 247 . . . . 5 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
77 elssuni 4607 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
7877adantl 473 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 (𝑘Gen‘𝐽))
793kgenuni 21515 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 = (𝑘Gen‘𝐽))
807, 79syl 17 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 = (𝑘Gen‘𝐽))
8178, 80sseqtr4d 3771 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 𝐽)
823isopn2 21009 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → (𝑥𝐽 ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
837, 81, 82syl2anc 696 . . . . 5 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥𝐽 ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
8476, 83mpbird 247 . . . 4 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥𝐽)
8584ex 449 . . 3 (𝐽 ∈ 1st𝜔 → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥𝐽))
8685ssrdv 3738 . 2 (𝐽 ∈ 1st𝜔 → (𝑘Gen‘𝐽) ⊆ 𝐽)
87 iskgen2 21524 . 2 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
881, 86, 87sylanbrc 701 1 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wex 1841  wcel 2127  Vcvv 3328  cdif 3700  cun 3701  cin 3702  wss 3703  {csn 4309   cuni 4576   class class class wbr 4792  ran crn 5255   Fn wfn 6032  wf 6033  cfv 6037  (class class class)co 6801  1c1 10100  cn 11183  t crest 16254  Topctop 20871  TopOnctopon 20888  Clsdccld 20993  clsccl 20995  𝑡clm 21203  Compccmp 21362  1st𝜔c1stc 21413  𝑘Genckgen 21509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-inf2 8699  ax-cc 9420  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-iin 4663  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7899  df-map 8013  df-pm 8014  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fi 8470  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-nn 11184  df-n0 11456  df-z 11541  df-uz 11851  df-fz 12491  df-rest 16256  df-topgen 16277  df-top 20872  df-topon 20889  df-bases 20923  df-cld 20996  df-ntr 20997  df-cls 20998  df-lm 21206  df-cmp 21363  df-1stc 21415  df-kgen 21510
This theorem is referenced by: (None)
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