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Theorem 1stckgen 21405
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 21294 . 2 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
2 difss 3770 . . . . . . . . . 10 ( 𝐽𝑥) ⊆ 𝐽
3 eqid 2651 . . . . . . . . . . 11 𝐽 = 𝐽
431stcelcls 21312 . . . . . . . . . 10 ((𝐽 ∈ 1st𝜔 ∧ ( 𝐽𝑥) ⊆ 𝐽) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
52, 4mpan2 707 . . . . . . . . 9 (𝐽 ∈ 1st𝜔 → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
65adantr 480 . . . . . . . 8 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘( 𝐽𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)))
71adantr 480 . . . . . . . . . . . . . 14 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 ∈ Top)
87adantr 480 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝐽 ∈ Top)
93toptopon 20770 . . . . . . . . . . . . 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 811 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓(⇝𝑡𝐽)𝑦)
12 lmcl 21149 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝑓(⇝𝑡𝐽)𝑦) → 𝑦 𝐽)
1310, 11, 12syl2anc 694 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 𝐽)
14 nnuz 11761 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
15 vex 3234 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
1615rnex 7142 . . . . . . . . . . . . . . . 16 ran 𝑓 ∈ V
17 snex 4938 . . . . . . . . . . . . . . . 16 {𝑦} ∈ V
1816, 17unex 6998 . . . . . . . . . . . . . . 15 (ran 𝑓 ∪ {𝑦}) ∈ V
19 resttop 21012 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ∈ V) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top)
208, 18, 19sylancl 695 . . . . . . . . . . . . . 14 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Top)
21 eqid 2651 . . . . . . . . . . . . . . 15 (𝐽t (ran 𝑓 ∪ {𝑦})) = (𝐽t (ran 𝑓 ∪ {𝑦}))
2221toptopon 20770 . . . . . . . . . . . . . 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 11446 . . . . . . . . . . . . 13 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 1 ∈ ℤ)
25 eqid 2651 . . . . . . . . . . . . . . 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 3810 . . . . . . . . . . . . . . . . 17 {𝑦} ⊆ (ran 𝑓 ∪ {𝑦})
28 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
2928snss 4348 . . . . . . . . . . . . . . . . 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 6083 . . . . . . . . . . . . . . . . . 18 (𝑓:ℕ⟶( 𝐽𝑥) → 𝑓 Fn ℕ)
3332ad2antrl 764 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓 Fn ℕ)
34 dffn3 6092 . . . . . . . . . . . . . . . . 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 3809 . . . . . . . . . . . . . . . 16 ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})
37 fss 6094 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ⟶ran 𝑓 ∧ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
3835, 36, 37sylancl 695 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
3925, 14, 26, 8, 31, 24, 38lmss 21150 . . . . . . . . . . . . . 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 6399 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ (ran 𝑓 ∪ {𝑦}))
42 simprl 809 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶( 𝐽𝑥))
4342ffvelrnda 6399 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ ( 𝐽𝑥))
4443eldifbd 3620 . . . . . . . . . . . . . 14 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → ¬ (𝑓𝑘) ∈ 𝑥)
4541, 44eldifd 3618 . . . . . . . . . . . . 13 ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
46 difin 3894 . . . . . . . . . . . . . . 15 ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥)
47 frn 6091 . . . . . . . . . . . . . . . . . . . 20 (𝑓:ℕ⟶( 𝐽𝑥) → ran 𝑓 ⊆ ( 𝐽𝑥))
4847ad2antrl 764 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ran 𝑓 ⊆ ( 𝐽𝑥))
4948difss2d 3773 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ran 𝑓 𝐽)
5013snssd 4372 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → {𝑦} ⊆ 𝐽)
5149, 50unssd 3822 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ⊆ 𝐽)
523restuni 21014 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ⊆ 𝐽) → (ran 𝑓 ∪ {𝑦}) = (𝐽t (ran 𝑓 ∪ {𝑦})))
538, 51, 52syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) = (𝐽t (ran 𝑓 ∪ {𝑦})))
5453difeq1d 3760 . . . . . . . . . . . . . . 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 2699 . . . . . . . . . . . . . 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 3838 . . . . . . . . . . . . . . . 16 ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) = (𝑥 ∩ (ran 𝑓 ∪ {𝑦}))
57 simplr 807 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑥 ∈ (𝑘Gen‘𝐽))
58 fss 6094 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ⟶( 𝐽𝑥) ∧ ( 𝐽𝑥) ⊆ 𝐽) → 𝑓:ℕ⟶ 𝐽)
5942, 2, 58sylancl 695 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑓:ℕ⟶ 𝐽)
6010, 59, 111stckgenlem 21404 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Comp)
61 kgeni 21388 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t (ran 𝑓 ∪ {𝑦})) ∈ Comp) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6257, 60, 61syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6356, 62syl5eqel 2734 . . . . . . . . . . . . . . 15 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) ∈ (𝐽t (ran 𝑓 ∪ {𝑦})))
6421opncld 20885 . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . 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 2730 . . . . . . . . . . . . 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 21155 . . . . . . . . . . . 12 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → 𝑦 ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
6867eldifbd 3620 . . . . . . . . . . 11 (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶( 𝐽𝑥) ∧ 𝑓(⇝𝑡𝐽)𝑦)) → ¬ 𝑦𝑥)
6913, 68eldifd 3618 . . . . . . . . . 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 1901 . . . . . . . 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 3642 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥))
743iscld4 20917 . . . . . . 7 ((𝐽 ∈ Top ∧ ( 𝐽𝑥) ⊆ 𝐽) → (( 𝐽𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘( 𝐽𝑥)) ⊆ ( 𝐽𝑥)))
757, 2, 74sylancl 695 . . . . . 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 4499 . . . . . . . 8 (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 (𝑘Gen‘𝐽))
7877adantl 481 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 (𝑘Gen‘𝐽))
793kgenuni 21390 . . . . . . . 8 (𝐽 ∈ Top → 𝐽 = (𝑘Gen‘𝐽))
807, 79syl 17 . . . . . . 7 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 = (𝑘Gen‘𝐽))
8178, 80sseqtr4d 3675 . . . . . 6 ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 𝐽)
823isopn2 20884 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → (𝑥𝐽 ↔ ( 𝐽𝑥) ∈ (Clsd‘𝐽)))
837, 81, 82syl2anc 694 . . . . 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 3642 . 2 (𝐽 ∈ 1st𝜔 → (𝑘Gen‘𝐽) ⊆ 𝐽)
87 iskgen2 21399 . 2 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
881, 86, 87sylanbrc 699 1 (𝐽 ∈ 1st𝜔 → 𝐽 ∈ ran 𝑘Gen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  {csn 4210   cuni 4468   class class class wbr 4685  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  1c1 9975  cn 11058  t crest 16128  Topctop 20746  TopOnctopon 20763  Clsdccld 20868  clsccl 20870  𝑡clm 21078  Compccmp 21237  1st𝜔c1stc 21288  𝑘Genckgen 21384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-lm 21081  df-cmp 21238  df-1stc 21290  df-kgen 21385
This theorem is referenced by: (None)
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