Theorem 1stckgen 22705

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 22594	. 2 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top)
2		difss 4066	. . . . . . . . . 10 ⊢ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽
3		eqid 2738	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	1stcelcls 22612	. . . . . . . . . 10 ⊢ ((𝐽 ∈ 1stω ∧ (∪ 𝐽 ∖ 𝑥) ⊆ ∪ 𝐽) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ↔ ∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)))
5	2, 4	mpan2 688	. . . . . . . . 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		toptopon2 22067	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
10	8, 9	sylib 217	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
11		simprr 770	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓(⇝𝑡‘𝐽)𝑦)
12		lmcl 22448	. . . . . . . . . . . 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 12621	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ≥‘1)
15		vex 3436	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
16	15	rnex 7759	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ∈ V
17		snex 5354	. . . . . . . . . . . . . . . 16 ⊢ {𝑦} ∈ V
18	16, 17	unex 7596	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑓 ∪ {𝑦}) ∈ V
19		resttop 22311	. . . . . . . . . . . . . . 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 22067	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Top ↔ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ (TopOn‘∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))))
22	20, 21	sylib 217	. . . . . . . . . . . . 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 12351	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 1 ∈ ℤ)
24		eqid 2738	. . . . . . . . . . . . . . 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 4107	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦})
27		vex 3436	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
28	27	snss 4719	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (ran 𝑓 ∪ {𝑦}) ↔ {𝑦} ⊆ (ran 𝑓 ∪ {𝑦}))
29	26, 28	mpbir 230	. . . . . . . . . . . . . . . 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 6600	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) → 𝑓 Fn ℕ)
32	31	ad2antrl 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓 Fn ℕ)
33		dffn3 6613	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn ℕ ↔ 𝑓:ℕ⟶ran 𝑓)
34	32, 33	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶ran 𝑓)
35		ssun1 4106	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑦})
36		fss 6617	. . . . . . . . . . . . . . . 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 22449	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 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 6961	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (ran 𝑓 ∪ {𝑦}))
41		simprl 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥))
42	41	ffvelrnda 6961	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ (∪ 𝐽 ∖ 𝑥))
43	42	eldifbd 3900	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → ¬ (𝑓‘𝑘) ∈ 𝑥)
44	40, 43	eldifd 3898	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
45		difin 4195	. . . . . . . . . . . . . . 15 ⊢ ((ran 𝑓 ∪ {𝑦}) ∖ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥)) = ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥)
46		frn 6607	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) → ran 𝑓 ⊆ (∪ 𝐽 ∖ 𝑥))
47	46	ad2antrl 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ran 𝑓 ⊆ (∪ 𝐽 ∖ 𝑥))
48	47	difss2d 4069	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ran 𝑓 ⊆ ∪ 𝐽)
49	13	snssd 4742	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → {𝑦} ⊆ ∪ 𝐽)
50	48, 49	unssd 4120	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (ran 𝑓 ∪ {𝑦}) ⊆ ∪ 𝐽)
51	3	restuni 22313	. . . . . . . . . . . . . . . . 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 4056	. . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . . . 16 ⊢ ((ran 𝑓 ∪ {𝑦}) ∩ 𝑥) = (𝑥 ∩ (ran 𝑓 ∪ {𝑦}))
56		simplr 766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑥 ∈ (𝑘Gen‘𝐽))
57		fss 6617	. . . . . . . . . . . . . . . . . . 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 22704	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) ∈ Comp)
60		kgeni 22688	. . . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . 16 ⊢ ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦})) = ∪ (𝐽 ↾t (ran 𝑓 ∪ {𝑦}))
64	63	opncld 22184	. . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . 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 22454	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ ((ran 𝑓 ∪ {𝑦}) ∖ 𝑥))
68	67	eldifbd 3900	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → ¬ 𝑦 ∈ 𝑥)
69	13, 68	eldifd 3898	. . . . . . . . . 10 ⊢ (((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) ∧ (𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦)) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥))
70	69	ex 413	. . . . . . . . 9 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
71	70	exlimdv 1936	. . . . . . . 8 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∃𝑓(𝑓:ℕ⟶(∪ 𝐽 ∖ 𝑥) ∧ 𝑓(⇝𝑡‘𝐽)𝑦) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
72	6, 71	sylbid 239	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑦 ∈ ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) → 𝑦 ∈ (∪ 𝐽 ∖ 𝑥)))
73	72	ssrdv 3927	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ((cls‘𝐽)‘(∪ 𝐽 ∖ 𝑥)) ⊆ (∪ 𝐽 ∖ 𝑥))
74	3	iscld4 22216	. . . . . . 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 256	. . . . 5 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (∪ 𝐽 ∖ 𝑥) ∈ (Clsd‘𝐽))
77		elssuni 4871	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
78	77	adantl 482	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ⊆ ∪ (𝑘Gen‘𝐽))
79	3	kgenuni 22690	. . . . . . . 8 ⊢ (𝐽 ∈ Top → ∪ 𝐽 = ∪ (𝑘Gen‘𝐽))
80	7, 79	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → ∪ 𝐽 = ∪ (𝑘Gen‘𝐽))
81	78, 80	sseqtrrd 3962	. . . . . 6 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ⊆ ∪ 𝐽)
82	3	isopn2 22183	. . . . . 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 256	. . . 4 ⊢ ((𝐽 ∈ 1stω ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ∈ 𝐽)
85	84	ex 413	. . 3 ⊢ (𝐽 ∈ 1stω → (𝑥 ∈ (𝑘Gen‘𝐽) → 𝑥 ∈ 𝐽))
86	85	ssrdv 3927	. 2 ⊢ (𝐽 ∈ 1stω → (𝑘Gen‘𝐽) ⊆ 𝐽)
87		iskgen2 22699	. 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 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  {csn 4561  ∪ cuni 4839   class class class wbr 5074  ran crn 5590   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  1c1 10872  ℕcn 11973   ↾_t crest 17131  Topctop 22042  TopOnctopon 22059  Clsdccld 22167  clsccl 22169  ⇝_𝑡clm 22377  Compccmp 22537  1^stωc1stc 22588  𝑘Genckgen 22684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cc 10191  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-lm 22380  df-cmp 22538  df-1stc 22590  df-kgen 22685

This theorem is referenced by: (None)