Theorem lly1stc 23000

Description: First-countability is a local property (unlike second-countability). (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
lly1stc	⊢ Locally 1stω = 1stω

Proof of Theorem lly1stc

Dummy variables 𝑗 𝑎 𝑛 𝑡 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		llytop 22976	. . . 4 ⊢ (𝑗 ∈ Locally 1stω → 𝑗 ∈ Top)
2		simprr 772	. . . . . . . . 9 ⊢ ((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) → (𝑗 ↾t 𝑢) ∈ 1stω)
3		simprl 770	. . . . . . . . . 10 ⊢ ((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) → 𝑥 ∈ 𝑢)
4	1	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) → 𝑗 ∈ Top)
5		elssuni 4942	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝑗 → 𝑢 ⊆ ∪ 𝑗)
6	5	ad2antlr 726	. . . . . . . . . . 11 ⊢ ((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) → 𝑢 ⊆ ∪ 𝑗)
7		eqid 2733	. . . . . . . . . . . 12 ⊢ ∪ 𝑗 = ∪ 𝑗
8	7	restuni 22666	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ⊆ ∪ 𝑗) → 𝑢 = ∪ (𝑗 ↾t 𝑢))
9	4, 6, 8	syl2anc 585	. . . . . . . . . 10 ⊢ ((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) → 𝑢 = ∪ (𝑗 ↾t 𝑢))
10	3, 9	eleqtrd 2836	. . . . . . . . 9 ⊢ ((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) → 𝑥 ∈ ∪ (𝑗 ↾t 𝑢))
11		eqid 2733	. . . . . . . . . 10 ⊢ ∪ (𝑗 ↾t 𝑢) = ∪ (𝑗 ↾t 𝑢)
12	11	1stcclb 22948	. . . . . . . . 9 ⊢ (((𝑗 ↾t 𝑢) ∈ 1stω ∧ 𝑥 ∈ ∪ (𝑗 ↾t 𝑢)) → ∃𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)(𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))
13	2, 10, 12	syl2anc 585	. . . . . . . 8 ⊢ ((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) → ∃𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)(𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))
14		elpwi 4610	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) → 𝑡 ⊆ (𝑗 ↾t 𝑢))
15	14	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) → 𝑡 ⊆ (𝑗 ↾t 𝑢))
16	15	sselda 3983	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → 𝑛 ∈ (𝑗 ↾t 𝑢))
17	4	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) → 𝑗 ∈ Top)
18		simpllr 775	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) → 𝑢 ∈ 𝑗)
19		restopn2 22681	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑛 ∈ (𝑗 ↾t 𝑢) ↔ (𝑛 ∈ 𝑗 ∧ 𝑛 ⊆ 𝑢)))
20	17, 18, 19	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) → (𝑛 ∈ (𝑗 ↾t 𝑢) ↔ (𝑛 ∈ 𝑗 ∧ 𝑛 ⊆ 𝑢)))
21	20	simplbda 501	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ (𝑗 ↾t 𝑢)) → 𝑛 ⊆ 𝑢)
22	16, 21	syldan 592	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → 𝑛 ⊆ 𝑢)
23		df-ss 3966	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ⊆ 𝑢 ↔ (𝑛 ∩ 𝑢) = 𝑛)
24	22, 23	sylib 217	. . . . . . . . . . . . . 14 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → (𝑛 ∩ 𝑢) = 𝑛)
25	20	simprbda 500	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ (𝑗 ↾t 𝑢)) → 𝑛 ∈ 𝑗)
26	16, 25	syldan 592	. . . . . . . . . . . . . 14 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → 𝑛 ∈ 𝑗)
27	24, 26	eqeltrd 2834	. . . . . . . . . . . . 13 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → (𝑛 ∩ 𝑢) ∈ 𝑗)
28		ineq1 4206	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑛 → (𝑎 ∩ 𝑢) = (𝑛 ∩ 𝑢))
29	28	cbvmptv 5262	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) = (𝑛 ∈ 𝑡 ↦ (𝑛 ∩ 𝑢))
30	27, 29	fmptd 7114	. . . . . . . . . . . 12 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) → (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)):𝑡⟶𝑗)
31	30	frnd 6726	. . . . . . . . . . 11 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) → ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ⊆ 𝑗)
32	31	adantrr 716	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) → ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ⊆ 𝑗)
33		vex 3479	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
34	33	elpw2 5346	. . . . . . . . . 10 ⊢ (ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ∈ 𝒫 𝑗 ↔ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ⊆ 𝑗)
35	32, 34	sylibr 233	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) → ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ∈ 𝒫 𝑗)
36		simprrl 780	. . . . . . . . . 10 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) → 𝑡 ≼ ω)
37		1stcrestlem 22956	. . . . . . . . . 10 ⊢ (𝑡 ≼ ω → ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ≼ ω)
38	36, 37	syl 17	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) → ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ≼ ω)
39		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ 𝑧)
40	3	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ 𝑢)
41	39, 40	elind 4195	. . . . . . . . . . . . 13 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ (𝑧 ∩ 𝑢))
42		eleq2 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝑥 ∈ 𝑣 ↔ 𝑥 ∈ (𝑧 ∩ 𝑢)))
43		sseq2 4009	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝑛 ⊆ 𝑣 ↔ 𝑛 ⊆ (𝑧 ∩ 𝑢)))
44	43	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ↔ (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢))))
45	44	rexbidv 3179	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ↔ ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢))))
46	42, 45	imbi12d 345	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)) ↔ (𝑥 ∈ (𝑧 ∩ 𝑢) → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢)))))
47		simprrr 781	. . . . . . . . . . . . . . 15 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) → ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)))
48	47	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)))
49	4	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → 𝑗 ∈ Top)
50		simpllr 775	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) → 𝑢 ∈ 𝑗)
51	50	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → 𝑢 ∈ 𝑗)
52		simprl 770	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → 𝑧 ∈ 𝑗)
53		elrestr 17374	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗 ∧ 𝑧 ∈ 𝑗) → (𝑧 ∩ 𝑢) ∈ (𝑗 ↾t 𝑢))
54	49, 51, 52, 53	syl3anc 1372	. . . . . . . . . . . . . 14 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → (𝑧 ∩ 𝑢) ∈ (𝑗 ↾t 𝑢))
55	46, 48, 54	rspcdva 3614	. . . . . . . . . . . . 13 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → (𝑥 ∈ (𝑧 ∩ 𝑢) → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢))))
56	41, 55	mpd 15	. . . . . . . . . . . 12 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢)))
57	3	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → 𝑥 ∈ 𝑢)
58		elin 3965	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑛 ∩ 𝑢) ↔ (𝑥 ∈ 𝑛 ∧ 𝑥 ∈ 𝑢))
59	58	simplbi2com 504	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑢 → (𝑥 ∈ 𝑛 → 𝑥 ∈ (𝑛 ∩ 𝑢)))
60	57, 59	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → (𝑥 ∈ 𝑛 → 𝑥 ∈ (𝑛 ∩ 𝑢)))
61	22	biantrud 533	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → (𝑛 ⊆ 𝑧 ↔ (𝑛 ⊆ 𝑧 ∧ 𝑛 ⊆ 𝑢)))
62		ssin 4231	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ⊆ 𝑧 ∧ 𝑛 ⊆ 𝑢) ↔ 𝑛 ⊆ (𝑧 ∩ 𝑢))
63	61, 62	bitrdi 287	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → (𝑛 ⊆ 𝑧 ↔ 𝑛 ⊆ (𝑧 ∩ 𝑢)))
64		ssinss1 4238	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ⊆ 𝑧 → (𝑛 ∩ 𝑢) ⊆ 𝑧)
65	63, 64	syl6bir 254	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → (𝑛 ⊆ (𝑧 ∩ 𝑢) → (𝑛 ∩ 𝑢) ⊆ 𝑧))
66	60, 65	anim12d 610	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → ((𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢)) → (𝑥 ∈ (𝑛 ∩ 𝑢) ∧ (𝑛 ∩ 𝑢) ⊆ 𝑧)))
67	66	reximdva 3169	. . . . . . . . . . . . . . 15 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) → (∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢)) → ∃𝑛 ∈ 𝑡 (𝑥 ∈ (𝑛 ∩ 𝑢) ∧ (𝑛 ∩ 𝑢) ⊆ 𝑧)))
68		vex 3479	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑛 ∈ V
69	68	inex1 5318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∩ 𝑢) ∈ V
70	69	rgenw 3066	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑛 ∈ 𝑡 (𝑛 ∩ 𝑢) ∈ V
71		eleq2 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑛 ∩ 𝑢) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ (𝑛 ∩ 𝑢)))
72		sseq1 4008	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑛 ∩ 𝑢) → (𝑤 ⊆ 𝑧 ↔ (𝑛 ∩ 𝑢) ⊆ 𝑧))
73	71, 72	anbi12d 632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑛 ∩ 𝑢) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝑥 ∈ (𝑛 ∩ 𝑢) ∧ (𝑛 ∩ 𝑢) ⊆ 𝑧)))
74	29, 73	rexrnmptw 7097	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ 𝑡 (𝑛 ∩ 𝑢) ∈ V → (∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑛 ∈ 𝑡 (𝑥 ∈ (𝑛 ∩ 𝑢) ∧ (𝑛 ∩ 𝑢) ⊆ 𝑧)))
75	70, 74	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑛 ∈ 𝑡 (𝑥 ∈ (𝑛 ∩ 𝑢) ∧ (𝑛 ∩ 𝑢) ⊆ 𝑧))
76	67, 75	imbitrrdi 251	. . . . . . . . . . . . . 14 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) → (∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢)) → ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
77	76	adantrr 716	. . . . . . . . . . . . 13 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) → (∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢)) → ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
78	77	adantr 482	. . . . . . . . . . . 12 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → (∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢)) → ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
79	56, 78	mpd 15	. . . . . . . . . . 11 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ (𝑧 ∈ 𝑗 ∧ 𝑥 ∈ 𝑧)) → ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))
80	79	expr 458	. . . . . . . . . 10 ⊢ ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) ∧ 𝑧 ∈ 𝑗) → (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
81	80	ralrimiva 3147	. . . . . . . . 9 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) → ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
82		breq1 5152	. . . . . . . . . . 11 ⊢ (𝑦 = ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) → (𝑦 ≼ ω ↔ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ≼ ω))
83		rexeq 3322	. . . . . . . . . . . . 13 ⊢ (𝑦 = ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) → (∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
84	83	imbi2d 341	. . . . . . . . . . . 12 ⊢ (𝑦 = ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) → ((𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
85	84	ralbidv 3178	. . . . . . . . . . 11 ⊢ (𝑦 = ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) → (∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
86	82, 85	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑦 = ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) → ((𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ↔ (ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))
87	86	rspcev 3613	. . . . . . . . 9 ⊢ ((ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ∈ 𝒫 𝑗 ∧ (ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
88	35, 38, 81, 87	syl12anc 836	. . . . . . . 8 ⊢ (((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣))))) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
89	13, 88	rexlimddv 3162	. . . . . . 7 ⊢ ((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
90	89	3adantr1 1170	. . . . . 6 ⊢ ((((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑢 ⊆ ∪ 𝑗 ∧ 𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
91		simpl 484	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) → 𝑗 ∈ Locally 1stω)
92	1	adantr 482	. . . . . . . 8 ⊢ ((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) → 𝑗 ∈ Top)
93	7	topopn 22408	. . . . . . . 8 ⊢ (𝑗 ∈ Top → ∪ 𝑗 ∈ 𝑗)
94	92, 93	syl 17	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) → ∪ 𝑗 ∈ 𝑗)
95		simpr 486	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) → 𝑥 ∈ ∪ 𝑗)
96		llyi 22978	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 1stω ∧ ∪ 𝑗 ∈ 𝑗 ∧ 𝑥 ∈ ∪ 𝑗) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ ∪ 𝑗 ∧ 𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω))
97	91, 94, 95, 96	syl3anc 1372	. . . . . 6 ⊢ ((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ ∪ 𝑗 ∧ 𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1stω))
98	90, 97	r19.29a 3163	. . . . 5 ⊢ ((𝑗 ∈ Locally 1stω ∧ 𝑥 ∈ ∪ 𝑗) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
99	98	ralrimiva 3147	. . . 4 ⊢ (𝑗 ∈ Locally 1stω → ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
100	7	is1stc2 22946	. . . 4 ⊢ (𝑗 ∈ 1stω ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))
101	1, 99, 100	sylanbrc 584	. . 3 ⊢ (𝑗 ∈ Locally 1stω → 𝑗 ∈ 1stω)
102	101	ssriv 3987	. 2 ⊢ Locally 1stω ⊆ 1stω
103		1stcrest 22957	. . . . 5 ⊢ ((𝑗 ∈ 1stω ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 1stω)
104	103	adantl 483	. . . 4 ⊢ ((⊤ ∧ (𝑗 ∈ 1stω ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 1stω)
105		1stctop 22947	. . . . . 6 ⊢ (𝑗 ∈ 1stω → 𝑗 ∈ Top)
106	105	ssriv 3987	. . . . 5 ⊢ 1stω ⊆ Top
107	106	a1i 11	. . . 4 ⊢ (⊤ → 1stω ⊆ Top)
108	104, 107	restlly 22987	. . 3 ⊢ (⊤ → 1stω ⊆ Locally 1stω)
109	108	mptru 1549	. 2 ⊢ 1stω ⊆ Locally 1stω
110	102, 109	eqssi 3999	1 ⊢ Locally 1stω = 1stω

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ⊤wtru 1543   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949  𝒫 cpw 4603  ∪ cuni 4909   class class class wbr 5149   ↦ cmpt 5232  ran crn 5678  (class class class)co 7409  ωcom 7855   ≼ cdom 8937   ↾_t crest 17366  Topctop 22395  1^stωc1stc 22941  Locally clly 22968

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9406  df-card 9934  df-acn 9937  df-rest 17368  df-topgen 17389  df-top 22396  df-topon 22413  df-bases 22449  df-1stc 22943  df-lly 22970

This theorem is referenced by:  dis1stc  23003