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Theorem 1stckgen 21637
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 21526 . 2 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
2 difss 3899 . . . . . . . . . 10 ( 𝐽𝑥) ⊆ 𝐽
3 eqid 2765 . . . . . . . . . . 11 𝐽 = 𝐽
431stcelcls 21544 . . . . . . . . . 10 ((𝐽 ∈ 1st𝜔 ∧ ( 𝐽𝑥) ⊆ 𝐽) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
52, 4mpan2 682 . . . . . . . . 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 21001 . . . . . . . . . . . . 13 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
108, 9sylib 209 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝐽 ∈ (TopOn‘ 𝐽))
11 simprr 789 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓(⇝𝑡𝐽)𝑦)
12 lmcl 21381 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 𝐽)
1310, 11, 12syl2anc 579 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 𝐽)
14 nnuz 11923 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
15 vex 3353 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
1615rnex 7298 . . . . . . . . . . . . . . . 16 ran 𝑓 ∈ V
17 snex 5064 . . . . . . . . . . . . . . . 16 {𝑦} ∈ V
1816, 17unex 7154 . . . . . . . . . . . . . . 15 (ran 𝑓 ∪ {𝑦}) ∈ V
19 resttop 21244 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ∈ V) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top)
208, 18, 19sylancl 580 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top)
21 eqid 2765 . . . . . . . . . . . . . . 15 (𝐽t (ran 𝑓 ∪ {𝑦})) = (𝐽t (ran 𝑓 ∪ {𝑦}))
2221toptopon 21001 . . . . . . . . . . . . . 14 ((𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top ↔ (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘ (𝐽t (ran 𝑓 ∪ {𝑦}))))
2320, 22sylib 209 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘ (𝐽t (ran 𝑓 ∪ {𝑦}))))
24 1zzd 11655 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 1 ∈ ℤ)
25 eqid 2765 . . . . . . . . . . . . . . 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 3939 . . . . . . . . . . . . . . . . 17 {𝑦} ⊆ (ran 𝑓 ∪ {𝑦})
28 vex 3353 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
2928snss 4470 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ran 𝑓 ∪ {𝑦}) ↔ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦}))
3027, 29mpbir 222 . . . . . . . . . . . . . . . 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 6223 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶( 𝐽𝑥) → 𝑓 Fn ℕ)
3332ad2antrl 719 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓 Fn ℕ)
34 dffn3 6234 . . . . . . . . . . . . . . . . 17 (𝑓 Fn ℕ ↔ 𝑓:ℕ⟶ran 𝑓)
3533, 34sylib 209 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶ran 𝑓)
36 ssun1 3938 . . . . . . . . . . . . . . . 16 ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})
37 fss 6236 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶ran 𝑓 ∧ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
3835, 36, 37sylancl 580 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
3925, 14, 26, 8, 31, 24, 38lmss 21382 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝑓(⇝𝑡𝐽)𝑦𝑓(⇝𝑡‘(𝐽t (ran 𝑓 ∪ {𝑦})))𝑦))
4011, 39mpbid 223 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓(⇝𝑡‘(𝐽t (ran 𝑓 ∪ {𝑦})))𝑦)
4138ffvelrnda 6549 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ (ran 𝑓 ∪ {𝑦}))
42 simprl 787 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶( 𝐽𝑥))
4342ffvelrnda 6549 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ ( 𝐽𝑥))
4443eldifbd 3745 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → ¬ (𝑓𝑘) ∈ 𝑥)
4541, 44eldifd 3743 . . . . . . . . . . . . 13 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
46 difin 4026 . . . . . . . . . . . . . . 15 ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥)
47 frn 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℕ⟶( 𝐽𝑥) → ran 𝑓 ⊆ ( 𝐽𝑥))
4847ad2antrl 719 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ran 𝑓 ⊆ ( 𝐽𝑥))
4948difss2d 3902 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ran 𝑓 𝐽)
5013snssd 4494 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → {𝑦} ⊆ 𝐽)
5149, 50unssd 3951 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ⊆ 𝐽)
523restuni 21246 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ⊆ 𝐽) → (ran 𝑓 ∪ {𝑦}) = (𝐽t (ran 𝑓 ∪ {𝑦})))
538, 51, 52syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) = (𝐽t (ran 𝑓 ∪ {𝑦})))
5453difeq1d 3889 . . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . 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 3967 . . . . . . . . . . . . . . . 16 ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) = (𝑥 ∩ (ran 𝑓 ∪ {𝑦}))
57 simplr 785 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑥 ∈ (𝑘Gen‘𝐽))
58 fss 6236 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ⟶( 𝐽𝑥) ∧ ( 𝐽𝑥) ⊆ 𝐽) → 𝑓:ℕ⟶ 𝐽)
5942, 2, 58sylancl 580 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶ 𝐽)
6010, 59, 111stckgenlem 21636 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Comp)
61 kgeni 21620 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Comp) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6257, 60, 61syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6356, 62syl5eqel 2848 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6421opncld 21117 . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . 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 21387 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
6867eldifbd 3745 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ¬ 𝑦𝑥)
6913, 68eldifd 3743 . . . . . . . . . 10 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ ( 𝐽𝑥))
7069ex 401 . . . . . . . . 9 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 ∈ ( 𝐽𝑥)))
7170exlimdv 2028 . . . . . . . 8 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 ∈ ( 𝐽𝑥)))
726, 71sylbid 231 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) → 𝑦 ∈ ( 𝐽𝑥)))
7372ssrdv 3767 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥))
743iscld4 21149 . . . . . . 7 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ⊆ 𝐽) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥)))
757, 2, 74sylancl 580 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥)))
7673, 75mpbird 248 . . . . 5 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ( 𝐽𝑥) ∈ (Clsd‘𝐽))
77 elssuni 4625 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
7877adantl 473 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 (𝑘Gen‘𝐽))
793kgenuni 21622 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 = (𝑘Gen‘𝐽))
807, 79syl 17 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 = (𝑘Gen‘𝐽))
8178, 80sseqtr4d 3802 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 𝐽)
823isopn2 21116 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → (𝑥𝐽 ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
837, 81, 82syl2anc 579 . . . . 5 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥𝐽 ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
8476, 83mpbird 248 . . . 4 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥𝐽)
8584ex 401 . . 3 (𝐽 ∈ 1st𝜔 → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥𝐽))
8685ssrdv 3767 . 2 (𝐽 ∈ 1st𝜔 → (𝑘Gen‘𝐽) ⊆ 𝐽)
87 iskgen2 21631 . 2 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
881, 86, 87sylanbrc 578 1 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  {csn 4334   cuni 4594   class class class wbr 4809  ran crn 5278   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  1c1 10190  cn 11274  t crest 16349  Topctop 20977  TopOnctopon 20994  Clsdccld 21100  clsccl 21102  𝑡clm 21310  Compccmp 21469  1st𝜔c1stc 21520  𝑘Genckgen 21616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cc 9510  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fi 8524  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-rest 16351  df-topgen 16372  df-top 20978  df-topon 20995  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105  df-lm 21313  df-cmp 21470  df-1stc 21522  df-kgen 21617
This theorem is referenced by: (None)
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