Theorem 1stckgen 23762

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 23650	. 2 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)
2		difss 4090	. . . . . . . . . 10 ⊢ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽
3		eqid 2765	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	1stcelcls 23669	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 1stω ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
5	2, 4	mpan2 704	. . . . . . . . 9 ⊢ (𝐽 ∈ 1stω → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
6	5	adantr 486	. . . . . . . 8 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
7	1	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝐽 ∈ Top)
8	7	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝐽 ∈ Top)
9		toptopon2 23125	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
10	8, 9	sylib 221	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
11		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓(⇝𝑡‘𝐽)𝑦)
12		lmcl 23504	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ ∪ 𝐽)
13	10, 11, 12	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ ∪ 𝐽)
14		nnuz 12917	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ≥‘1)
15		vex 3461	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
16	15	rnex 7913	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ∈ V
17		vsnex 5408	. . . . . . . . . . . . . . . 16 ⊢ {𝑦} ∈ V
18	16, 17	unex 7752	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑓 ∪ {𝑦}) ∈ V
19		resttop 23367	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ∈ V) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top)
20	8, 18, 19	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top)
21		toptopon2 23125	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top ↔ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))))
22	20, 21	sylib 221	. . . . . . . . . . . . 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 12640	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 1 ∈ ℤ)
24		eqid 2765	. . . . . . . . . . . . . . 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 4132	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦})
27		vex 3461	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
28	27	snss 4752	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ran 𝑓 ∪ {𝑦}) ↔ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦}))
29	26, 28	mpbir 234	. . . . . . . . . . . . . . . 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 6709	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) → 𝑓 Fn ℕ)
32	31	ad2antrl 741	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓 Fn ℕ)
33		dffn3 6722	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn ℕ ↔ 𝑓:ℕ⟶ran 𝑓)
34	32, 33	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶ran 𝑓)
35		ssun1 4131	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})
36		fss 6726	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ⟶ran 𝑓 ∧ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})) → 𝑓:ℕ⟶(ran 𝑓 ∪ {𝑦}))
37	34, 35, 36	sylancl 598	. . . . . . . . . . . . . . 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 23505	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 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 7083	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (ran 𝑓 ∪ {𝑦}))
41		simprl 783	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥))
42	41	ffvelcdmda 7083	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (∪ 𝐽 ∖ 𝑥))
43	42	eldifbd 3919	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → ¬ (𝑓‘𝑘) ∈ 𝑥)
44	40, 43	eldifd 3917	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
45		difin 4225	. . . . . . . . . . . . . . 15 ⊢ ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥)
46		frn 6717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) → ran 𝑓 ⊆ (∪ 𝐽 ∖ 𝑥))
47	46	ad2antrl 741	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ran 𝑓 ⊆ (∪ 𝐽 ∖ 𝑥))
48	47	difss2d 4093	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ran 𝑓 ⊆ ∪ 𝐽)
49	13	snssd 4754	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → {𝑦} ⊆ ∪ 𝐽)
50	48, 49	unssd 4145	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ⊆ ∪ 𝐽)
51	3	restuni 23369	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ (ran 𝑓 ∪ {𝑦}) ⊆ ∪ 𝐽) → (ran 𝑓 ∪ {𝑦}) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
52	8, 50, 51	syl2anc 596	. . . . . . . . . . . . . . . 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 4080	. . . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . . 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 4162	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) = (𝑥 ∩ (ran 𝑓 ∪ {𝑦}))
56		simplr 781	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑥 ∈ (𝑘Gen‘𝐽))
57		fss 6726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → 𝑓:ℕ⟶∪ 𝐽)
58	41, 2, 57	sylancl 598	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶∪ 𝐽)
59	10, 58, 11	1stckgenlem 23761	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Comp)
60		kgeni 23745	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Comp) → (𝑥 ∩ (ran 𝑓 ∪ {𝑦})) ∈ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})))
61	56, 59, 60	syl2anc 596	. . . . . . . . . . . . . . . 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 2869	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . 16 ⊢ ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))
64	63	opncld 23240	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . 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 23510	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
68	67	eldifbd 3919	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ¬ 𝑦 ∈ 𝑥)
69	13, 68	eldifd 3917	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥))
70	69	ex 418	. . . . . . . . 9 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
71	70	exlimdv 1966	. . . . . . . 8 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
72	6, 71	sylbid 243	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
73	72	ssrdv 3944	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥))
74	3	iscld4 23272	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥)))
75	7, 2, 74	sylancl 598	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥)))
76	73, 75	mpbird 260	. . . . 5 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
77		elssuni 4906	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
78	77	adantl 487	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
79	3	kgenuni 23747	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ∪ 𝐽 = ∪ (𝑘Gen‘𝐽))
80	7, 79	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ∪ 𝐽 = ∪ (𝑘Gen‘𝐽))
81	78, 80	sseqtrrd 3975	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ⊆ ∪ 𝐽)
82	3	isopn2 23239	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
83	7, 81, 82	syl2anc 596	. . . . 5 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥 ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽)))
84	76, 83	mpbird 260	. . . 4 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ∈ 𝐽)
85	84	ex 418	. . 3 ⊢ (𝐽 ∈ 1stω → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ 𝐽))
86	85	ssrdv 3944	. 2 ⊢ (𝐽 ∈ 1stω → (𝑘Gen‘𝐽) ⊆ 𝐽)
87		iskgen2 23756	. 2 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
88	1, 86, 87	sylanbrc 595	1 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ ran 𝑘Gen)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3457   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  {csn 4591  ∪ cuni 4874   class class class wbr 5111  ran crn 5664   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  1c1 11116  ℕcn 12248   ↾_t crest 17495  Topctop 23100  TopOnctopon 23117  Clsdccld 23223  clsccl 23225  ⇝_𝑡clm 23433  Compccmp 23593  1^stωc1stc 23644  𝑘Genckgen 23741

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cc 10434  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-pm 8833  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fi 9378  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-n0 12520  df-z 12607  df-uz 12879  df-fz 13552  df-rest 17497  df-topgen 17518  df-top 23101  df-topon 23118  df-bases 23153  df-cld 23226  df-ntr 23227  df-cls 23228  df-lm 23436  df-cmp 23594  df-1stc 23646  df-kgen 23742

This theorem is used by: (None)