Theorem 1stckgen 23417

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 23306	. 2 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)
2		difss 4095	. . . . . . . . . 10 ⊢ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽
3		eqid 2729	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	1stcelcls 23324	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 1stω ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
5	2, 4	mpan2 691	. . . . . . . . 9 ⊢ (𝐽 ∈ 1stω → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
6	5	adantr 480	. . . . . . . 8 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
7	1	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 ∈ Top)
8	7	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝐽 ∈ Top)
9		toptopon2 22781	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
10	8, 9	sylib 218	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
11		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓(⇝𝑡‘𝐽)𝑦)
12		lmcl 23160	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ ∪ 𝐽)
13	10, 11, 12	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ ∪ 𝐽)
14		nnuz 12812	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ≥‘1)
15		vex 3448	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
16	15	rnex 7866	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ∈ V
17		vsnex 5384	. . . . . . . . . . . . . . . 16 ⊢ {𝑦} ∈ V
18	16, 17	unex 7700	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑓 ∪ {𝑦}) ∈ V
19		resttop 23023	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ∈ V) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top)
20	8, 18, 19	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top)
21		toptopon2 22781	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top ↔ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))))
22	20, 21	sylib 218	. . . . . . . . . . . . 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}))))
23		1zzd 12540	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 1 ∈ ℤ)
24		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) = (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))
25	18	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ∈ V)
26		ssun2 4138	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦})
27		vex 3448	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
28	27	snss 4745	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ran 𝑓 ∪ {𝑦}) ↔ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦}))
29	26, 28	mpbir 231	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ (ran 𝑓 ∪ {𝑦})
30	29	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ (ran 𝑓 ∪ {𝑦}))
31		ffn 6670	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) → 𝑓 Fn ℕ)
32	31	ad2antrl 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓 Fn ℕ)
33		dffn3 6682	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn ℕ ↔ 𝑓:ℕ⟶ran 𝑓)
34	32, 33	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶ran 𝑓)
35		ssun1 4137	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})
36		fss 6686	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶ran 𝑓 ∧ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
37	34, 35, 36	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
38	24, 14, 25, 8, 30, 23, 37	lmss 23161	. . . . . . . . . . . . . 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))
39	11, 38	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓(⇝𝑡‘(𝐽 ↾t (ran 𝑓 ∪ {𝑦})))𝑦)
40	37	ffvelcdmda 7038	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (ran 𝑓 ∪ {𝑦}))
41		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥))
42	41	ffvelcdmda 7038	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (∪ 𝐽 ∖ 𝑥))
43	42	eldifbd 3924	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → ¬ (𝑓‘𝑘) ∈ 𝑥)
44	40, 43	eldifd 3922	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
45		difin 4231	. . . . . . . . . . . . . . 15 ⊢ ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥)
46		frn 6677	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) → ran 𝑓 ⊆ (∪ 𝐽 ∖ 𝑥))
47	46	ad2antrl 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ran 𝑓 ⊆ (∪ 𝐽 ∖ 𝑥))
48	47	difss2d 4098	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ran 𝑓 ⊆ ∪ 𝐽)
49	13	snssd 4769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → {𝑦} ⊆ ∪ 𝐽)
50	48, 49	unssd 4151	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ⊆ ∪ 𝐽)
51	3	restuni 23025	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ⊆ ∪ 𝐽) → (ran 𝑓 ∪ {𝑦}) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
52	8, 50, 51	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
53	52	difeq1d 4084	. . . . . . . . . . . . . . 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)))
54	45, 53	eqtr3id 2778	. . . . . . . . . . . . . 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)))
55		incom 4168	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) = (𝑥 ∩ (ran 𝑓 ∪ {𝑦}))
56		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑥 ∈ (𝑘Gen‘𝐽))
57		fss 6686	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → 𝑓:ℕ⟶∪ 𝐽)
58	41, 2, 57	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶∪ 𝐽)
59	10, 58, 11	1stckgenlem 23416	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Comp)
60		kgeni 23400	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Comp) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
61	56, 59, 60	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
62	55, 61	eqeltrid 2832	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) ∈ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
63		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))
64	63	opncld 22896	. . . . . . . . . . . . . . 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, 62, 64	syl2anc 584	. . . . . . . . . . . . . 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	54, 65	eqeltrd 2828	. . . . . . . . . . . . 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, 22, 23, 39, 44, 66	lmcld 23166	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
68	67	eldifbd 3924	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ¬ 𝑦 ∈ 𝑥)
69	13, 68	eldifd 3922	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥))
70	69	ex 412	. . . . . . . . 9 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
71	70	exlimdv 1933	. . . . . . . 8 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
72	6, 71	sylbid 240	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
73	72	ssrdv 3949	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥))
74	3	iscld4 22928	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥)))
75	7, 2, 74	sylancl 586	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥)))
76	73, 75	mpbird 257	. . . . 5 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
77		elssuni 4897	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
78	77	adantl 481	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
79	3	kgenuni 23402	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ∪ 𝐽 = ∪ (𝑘Gen‘𝐽))
80	7, 79	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ∪ 𝐽 = ∪ (𝑘Gen‘𝐽))
81	78, 80	sseqtrrd 3981	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ⊆ ∪ 𝐽)
82	3	isopn2 22895	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
83	7, 81, 82	syl2anc 584	. . . . 5 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥 ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
84	76, 83	mpbird 257	. . . 4 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ∈ 𝐽)
85	84	ex 412	. . 3 ⊢ (𝐽 ∈ 1stω → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ 𝐽))
86	85	ssrdv 3949	. 2 ⊢ (𝐽 ∈ 1stω → (𝑘Gen‘𝐽) ⊆ 𝐽)
87		iskgen2 23411	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
88	1, 86, 87	sylanbrc 583	1 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ ran 𝑘Gen)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  {csn 4585  ∪ cuni 4867   class class class wbr 5102  ran crn 5632   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  1c1 11045  ℕcn 12162   ↾_t crest 17359  Topctop 22756  TopOnctopon 22773  Clsdccld 22879  clsccl 22881  ⇝_𝑡clm 23089  Compccmp 23249  1^stωc1stc 23300  𝑘Genckgen 23396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-inf2 9570  ax-cc 10364  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-pm 8779  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fi 9338  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-n0 12419  df-z 12506  df-uz 12770  df-fz 13445  df-rest 17361  df-topgen 17382  df-top 22757  df-topon 22774  df-bases 22809  df-cld 22882  df-ntr 22883  df-cls 22884  df-lm 23092  df-cmp 23250  df-1stc 23302  df-kgen 23397

This theorem is referenced by: (None)