Theorem 1stckgen 22156

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 22045	. 2 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)
2		difss 4108	. . . . . . . . . 10 ⊢ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽
3		eqid 2821	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	1stcelcls 22063	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 1stω ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
5	2, 4	mpan2 689	. . . . . . . . 9 ⊢ (𝐽 ∈ 1stω → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
6	5	adantr 483	. . . . . . . 8 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
7	1	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 ∈ Top)
8	7	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝐽 ∈ Top)
9		toptopon2 21520	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
10	8, 9	sylib 220	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
11		simprr 771	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓(⇝𝑡‘𝐽)𝑦)
12		lmcl 21899	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ ∪ 𝐽)
13	10, 11, 12	syl2anc 586	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ ∪ 𝐽)
14		nnuz 12275	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ≥‘1)
15		vex 3498	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
16	15	rnex 7611	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ∈ V
17		snex 5324	. . . . . . . . . . . . . . . 16 ⊢ {𝑦} ∈ V
18	16, 17	unex 7463	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑓 ∪ {𝑦}) ∈ V
19		resttop 21762	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ∈ V) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top)
20	8, 18, 19	sylancl 588	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top)
21		toptopon2 21520	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top ↔ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))))
22	20, 21	sylib 220	. . . . . . . . . . . . 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 12007	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 1 ∈ ℤ)
24		eqid 2821	. . . . . . . . . . . . . . 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 4149	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦})
27		vex 3498	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
28	27	snss 4712	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ran 𝑓 ∪ {𝑦}) ↔ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦}))
29	26, 28	mpbir 233	. . . . . . . . . . . . . . . 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 6509	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) → 𝑓 Fn ℕ)
32	31	ad2antrl 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓 Fn ℕ)
33		dffn3 6520	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn ℕ ↔ 𝑓:ℕ⟶ran 𝑓)
34	32, 33	sylib 220	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶ran 𝑓)
35		ssun1 4148	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})
36		fss 6522	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶ran 𝑓 ∧ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
37	34, 35, 36	sylancl 588	. . . . . . . . . . . . . . 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 21900	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓(⇝𝑡‘(𝐽 ↾t (ran 𝑓 ∪ {𝑦})))𝑦)
40	37	ffvelrnda 6846	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (ran 𝑓 ∪ {𝑦}))
41		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥))
42	41	ffvelrnda 6846	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (∪ 𝐽 ∖ 𝑥))
43	42	eldifbd 3949	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → ¬ (𝑓‘𝑘) ∈ 𝑥)
44	40, 43	eldifd 3947	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
45		difin 4238	. . . . . . . . . . . . . . 15 ⊢ ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥)
46		frn 6515	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) → ran 𝑓 ⊆ (∪ 𝐽 ∖ 𝑥))
47	46	ad2antrl 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ran 𝑓 ⊆ (∪ 𝐽 ∖ 𝑥))
48	47	difss2d 4111	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ran 𝑓 ⊆ ∪ 𝐽)
49	13	snssd 4736	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → {𝑦} ⊆ ∪ 𝐽)
50	48, 49	unssd 4162	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ⊆ ∪ 𝐽)
51	3	restuni 21764	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ⊆ ∪ 𝐽) → (ran 𝑓 ∪ {𝑦}) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
52	8, 50, 51	syl2anc 586	. . . . . . . . . . . . . . . 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 4098	. . . . . . . . . . . . . . 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	syl5eqr 2870	. . . . . . . . . . . . . 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 4178	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) = (𝑥 ∩ (ran 𝑓 ∪ {𝑦}))
56		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑥 ∈ (𝑘Gen‘𝐽))
57		fss 6522	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → 𝑓:ℕ⟶∪ 𝐽)
58	41, 2, 57	sylancl 588	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶∪ 𝐽)
59	10, 58, 11	1stckgenlem 22155	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Comp)
60		kgeni 22139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Comp) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
61	56, 59, 60	syl2anc 586	. . . . . . . . . . . . . . . 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 2917	. . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . 16 ⊢ ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))
64	63	opncld 21635	. . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . 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 2913	. . . . . . . . . . . . 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 21905	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
68	67	eldifbd 3949	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ¬ 𝑦 ∈ 𝑥)
69	13, 68	eldifd 3947	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥))
70	69	ex 415	. . . . . . . . 9 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
71	70	exlimdv 1930	. . . . . . . 8 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
72	6, 71	sylbid 242	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
73	72	ssrdv 3973	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥))
74	3	iscld4 21667	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥)))
75	7, 2, 74	sylancl 588	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥)))
76	73, 75	mpbird 259	. . . . 5 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
77		elssuni 4861	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
78	77	adantl 484	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
79	3	kgenuni 22141	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ∪ 𝐽 = ∪ (𝑘Gen‘𝐽))
80	7, 79	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ∪ 𝐽 = ∪ (𝑘Gen‘𝐽))
81	78, 80	sseqtrrd 4008	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ⊆ ∪ 𝐽)
82	3	isopn2 21634	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
83	7, 81, 82	syl2anc 586	. . . . 5 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥 ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
84	76, 83	mpbird 259	. . . 4 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ∈ 𝐽)
85	84	ex 415	. . 3 ⊢ (𝐽 ∈ 1stω → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ 𝐽))
86	85	ssrdv 3973	. 2 ⊢ (𝐽 ∈ 1stω → (𝑘Gen‘𝐽) ⊆ 𝐽)
87		iskgen2 22150	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
88	1, 86, 87	sylanbrc 585	1 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ ran 𝑘Gen)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3495   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935   ⊆ wss 3936  {csn 4561  ∪ cuni 4832   class class class wbr 5059  ran crn 5551   Fn wfn 6345  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150  1c1 10532  ℕcn 11632   ↾_t crest 16688  Topctop 21495  TopOnctopon 21512  Clsdccld 21618  clsccl 21620  ⇝_𝑡clm 21828  Compccmp 21988  1^stωc1stc 22039  𝑘Genckgen 22135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098  ax-cc 9851  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-rest 16690  df-topgen 16711  df-top 21496  df-topon 21513  df-bases 21548  df-cld 21621  df-ntr 21622  df-cls 21623  df-lm 21831  df-cmp 21989  df-1stc 22041  df-kgen 22136

This theorem is referenced by: (None)