Theorem 1stckgen 21357

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.

Step	Hyp	Ref	Expression
1		1stctop 21246	. 2 ⊢ (𝐽 ∈ 1st𝜔 → 𝐽 ∈ Top)
2		difss 3737	. . . . . . . . . 10 ⊢ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽
3		eqid 2622	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	1stcelcls 21264	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 1st𝜔 ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
5	2, 4	mpan2 707	. . . . . . . . 9 ⊢ (𝐽 ∈ 1st𝜔 → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
6	5	adantr 481	. . . . . . . 8 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
7	1	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 ∈ Top)
8	7	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝐽 ∈ Top)
9	3	toptopon 20722	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
10	8, 9	sylib 208	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
11		simprr 796	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓(⇝𝑡‘𝐽)𝑦)
12		lmcl 21101	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ ∪ 𝐽)
13	10, 11, 12	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ ∪ 𝐽)
14		nnuz 11723	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ≥‘1)
15		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
16	15	rnex 7100	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ∈ V
17		snex 4908	. . . . . . . . . . . . . . . 16 ⊢ {𝑦} ∈ V
18	16, 17	unex 6956	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑓 ∪ {𝑦}) ∈ V
19		resttop 20964	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ∈ V) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top)
20	8, 18, 19	sylancl 694	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top)
21		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))
22	21	toptopon 20722	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top ↔ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))))
23	20, 22	sylib 208	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))))
24		1zzd 11408	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 1 ∈ ℤ)
25		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) = (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))
26	18	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ∈ V)
27		ssun2 3777	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦})
28		vex 3203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
29	28	snss 4316	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ran 𝑓 ∪ {𝑦}) ↔ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦}))
30	27, 29	mpbir 221	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ (ran 𝑓 ∪ {𝑦})
31	30	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ (ran 𝑓 ∪ {𝑦}))
32		ffn 6045	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) → 𝑓 Fn ℕ)
33	32	ad2antrl 764	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓 Fn ℕ)
34		dffn3 6054	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn ℕ ↔ 𝑓:ℕ⟶ran 𝑓)
35	33, 34	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶ran 𝑓)
36		ssun1 3776	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})
37		fss 6056	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶ran 𝑓 ∧ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
38	35, 36, 37	sylancl 694	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
39	25, 14, 26, 8, 31, 24, 38	lmss 21102	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝑓(⇝𝑡‘𝐽)𝑦 ↔ 𝑓(⇝𝑡‘(𝐽 ↾t (ran 𝑓 ∪ {𝑦})))𝑦))
40	11, 39	mpbid 222	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓(⇝𝑡‘(𝐽 ↾t (ran 𝑓 ∪ {𝑦})))𝑦)
41	38	ffvelrnda 6359	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (ran 𝑓 ∪ {𝑦}))
42		simprl 794	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥))
43	42	ffvelrnda 6359	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (∪ 𝐽 ∖ 𝑥))
44	43	eldifbd 3587	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → ¬ (𝑓‘𝑘) ∈ 𝑥)
45	41, 44	eldifd 3585	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
46		difin 3861	. . . . . . . . . . . . . . 15 ⊢ ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥)
47		frn 6053	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) → ran 𝑓 ⊆ (∪ 𝐽 ∖ 𝑥))
48	47	ad2antrl 764	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ran 𝑓 ⊆ (∪ 𝐽 ∖ 𝑥))
49	48	difss2d 3740	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ran 𝑓 ⊆ ∪ 𝐽)
50	13	snssd 4340	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → {𝑦} ⊆ ∪ 𝐽)
51	49, 50	unssd 3789	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ⊆ ∪ 𝐽)
52	3	restuni 20966	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ⊆ ∪ 𝐽) → (ran 𝑓 ∪ {𝑦}) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
53	8, 51, 52	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
54	53	difeq1d 3727	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = (∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)))
55	46, 54	syl5eqr 2670	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥) = (∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)))
56		incom 3805	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) = (𝑥 ∩ (ran 𝑓 ∪ {𝑦}))
57		simplr 792	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑥 ∈ (𝑘Gen‘𝐽))
58		fss 6056	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → 𝑓:ℕ⟶∪ 𝐽)
59	42, 2, 58	sylancl 694	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶∪ 𝐽)
60	10, 59, 11	1stckgenlem 21356	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Comp)
61		kgeni 21340	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Comp) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
62	57, 60, 61	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
63	56, 62	syl5eqel 2705	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) ∈ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
64	21	opncld 20837	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top ∧ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) ∈ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))) → (∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) ∈ (Clsd‘(𝐽 ↾t (ran 𝑓 ∪ {𝑦}))))
65	20, 63, 64	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) ∈ (Clsd‘(𝐽 ↾t (ran 𝑓 ∪ {𝑦}))))
66	55, 65	eqeltrd 2701	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥) ∈ (Clsd‘(𝐽 ↾t (ran 𝑓 ∪ {𝑦}))))
67	14, 23, 24, 40, 45, 66	lmcld 21107	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
68	67	eldifbd 3587	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ¬ 𝑦 ∈ 𝑥)
69	13, 68	eldifd 3585	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥))
70	69	ex 450	. . . . . . . . 9 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
71	70	exlimdv 1861	. . . . . . . 8 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
72	6, 71	sylbid 230	. . . . . . 7 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
73	72	ssrdv 3609	. . . . . 6 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥))
74	3	iscld4 20869	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥)))
75	7, 2, 74	sylancl 694	. . . . . 6 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥)))
76	73, 75	mpbird 247	. . . . 5 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
77		elssuni 4467	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
78	77	adantl 482	. . . . . . 7 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
79	3	kgenuni 21342	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ∪ 𝐽 = ∪ (𝑘Gen‘𝐽))
80	7, 79	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ∪ 𝐽 = ∪ (𝑘Gen‘𝐽))
81	78, 80	sseqtr4d 3642	. . . . . 6 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ⊆ ∪ 𝐽)
82	3	isopn2 20836	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
83	7, 81, 82	syl2anc 693	. . . . 5 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥 ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
84	76, 83	mpbird 247	. . . 4 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ∈ 𝐽)
85	84	ex 450	. . 3 ⊢ (𝐽 ∈ 1st𝜔 → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ 𝐽))
86	85	ssrdv 3609	. 2 ⊢ (𝐽 ∈ 1st𝜔 → (𝑘Gen‘𝐽) ⊆ 𝐽)
87		iskgen2 21351	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
88	1, 86, 87	sylanbrc 698	1 ⊢ (𝐽 ∈ 1st𝜔 → 𝐽 ∈ ran 𝑘Gen)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  {csn 4177  ∪ cuni 4436   class class class wbr 4653  ran crn 5115   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1c1 9937  ℕcn 11020   ↾_t crest 16081  Topctop 20698  TopOnctopon 20715  Clsdccld 20820  clsccl 20822  ⇝_𝑡clm 21030  Compccmp 21189  1^st𝜔c1stc 21240  𝑘Genckgen 21336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-lm 21033  df-cmp 21190  df-1stc 21242  df-kgen 21337

This theorem is referenced by: (None)