Theorem lly1stc 21109

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 21085	. . . 4 ⊢ (𝑗 ∈ Locally 1st𝜔 → 𝑗 ∈ Top)
2		simprr 792	. . . . . . . . 9 ⊢ ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔)) → (𝑗 ↾t 𝑢) ∈ 1st𝜔)
3		simprl 790	. . . . . . . . . 10 ⊢ ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔)) → 𝑥 ∈ 𝑢)
4	1	ad3antrrr 762	. . . . . . . . . . 11 ⊢ ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔)) → 𝑗 ∈ Top)
5		elssuni 4403	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝑗 → 𝑢 ⊆ ∪ 𝑗)
6	5	ad2antlr 759	. . . . . . . . . . 11 ⊢ ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔)) → 𝑢 ⊆ ∪ 𝑗)
7		eqid 2610	. . . . . . . . . . . 12 ⊢ ∪ 𝑗 = ∪ 𝑗
8	7	restuni 20776	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ⊆ ∪ 𝑗) → 𝑢 = ∪ (𝑗 ↾t 𝑢))
9	4, 6, 8	syl2anc 691	. . . . . . . . . 10 ⊢ ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔)) → 𝑢 = ∪ (𝑗 ↾t 𝑢))
10	3, 9	eleqtrd 2690	. . . . . . . . 9 ⊢ ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔)) → 𝑥 ∈ ∪ (𝑗 ↾t 𝑢))
11		eqid 2610	. . . . . . . . . 10 ⊢ ∪ (𝑗 ↾t 𝑢) = ∪ (𝑗 ↾t 𝑢)
12	11	1stcclb 21057	. . . . . . . . 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 691	. . . . . . . 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 4117	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢) → 𝑡 ⊆ (𝑗 ↾t 𝑢))
15	14	adantl 481	. . . . . . . . . . . . . . . . 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 3568	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) → 𝑗 ∈ Top)
18		simpllr 795	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) → 𝑢 ∈ 𝑗)
19		restopn2 20791	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗) → (𝑛 ∈ (𝑗 ↾t 𝑢) ↔ (𝑛 ∈ 𝑗 ∧ 𝑛 ⊆ 𝑢)))
20	17, 18, 19	syl2anc 691	. . . . . . . . . . . . . . . . 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 652	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 3554	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ⊆ 𝑢 ↔ (𝑛 ∩ 𝑢) = 𝑛)
24	22, 23	sylib 207	. . . . . . . . . . . . . 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 651	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 2688	. . . . . . . . . . . . 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 3769	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑛 → (𝑎 ∩ 𝑢) = (𝑛 ∩ 𝑢))
29	28	cbvmptv 4678	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) = (𝑛 ∈ 𝑡 ↦ (𝑛 ∩ 𝑢))
30	27, 29	fmptd 6292	. . . . . . . . . . . 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		frn 5966	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)):𝑡⟶𝑗 → ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ⊆ 𝑗)
32	30, 31	syl 17	. . . . . . . . . . 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)
33	32	adantrr 749	. . . . . . . . . 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)
34		vex 3176	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
35	34	elpw2 4755	. . . . . . . . . 10 ⊢ (ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ∈ 𝒫 𝑗 ↔ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ⊆ 𝑗)
36	33, 35	sylibr 223	. . . . . . . . 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)
37		simprrl 800	. . . . . . . . . 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)
38		1stcrestlem 21065	. . . . . . . . . 10 ⊢ (𝑡 ≼ ω → ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ≼ ω)
39	37, 38	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)
40	4	ad2antrr 758	. . . . . . . . . . . . . 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)) → 𝑗 ∈ Top)
41		simpllr 795	. . . . . . . . . . . . . . 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))))) → 𝑢 ∈ 𝑗)
42	41	adantr 480	. . . . . . . . . . . . . 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)) → 𝑢 ∈ 𝑗)
43		simprl 790	. . . . . . . . . . . . . 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 ∈ 𝑗)
44		elrestr 15912	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ Top ∧ 𝑢 ∈ 𝑗 ∧ 𝑧 ∈ 𝑗) → (𝑧 ∩ 𝑢) ∈ (𝑗 ↾t 𝑢))
45	40, 42, 43, 44	syl3anc 1318	. . . . . . . . . . . . 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)) → (𝑧 ∩ 𝑢) ∈ (𝑗 ↾t 𝑢))
46		simprrr 801	. . . . . . . . . . . . . 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))))) → ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)))
47	46	adantr 480	. . . . . . . . . . . . 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)) → ∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)))
48		simprr 792	. . . . . . . . . . . . . 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)
49	3	ad2antrr 758	. . . . . . . . . . . . . 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)
50	48, 49	elind 3760	. . . . . . . . . . . . 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))
51		eleq2 2677	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝑥 ∈ 𝑣 ↔ 𝑥 ∈ (𝑧 ∩ 𝑢)))
52		sseq2 3590	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (𝑛 ⊆ 𝑣 ↔ 𝑛 ⊆ (𝑧 ∩ 𝑢)))
53	52	anbi2d 736	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ↔ (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢))))
54	53	rexbidv 3034	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → (∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣) ↔ ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢))))
55	51, 54	imbi12d 333	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 ∩ 𝑢) → ((𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)) ↔ (𝑥 ∈ (𝑧 ∩ 𝑢) → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢)))))
56	55	rspcv 3278	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑢) ∈ (𝑗 ↾t 𝑢) → (∀𝑣 ∈ (𝑗 ↾t 𝑢)(𝑥 ∈ 𝑣 → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ 𝑣)) → (𝑥 ∈ (𝑧 ∩ 𝑢) → ∃𝑛 ∈ 𝑡 (𝑥 ∈ 𝑛 ∧ 𝑛 ⊆ (𝑧 ∩ 𝑢)))))
57	45, 47, 50, 56	syl3c 64	. . . . . . . . . . . 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)))
58	3	ad2antrr 758	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑢 ∈ 𝑗) ∧ (𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗 ↾t 𝑢)) ∧ 𝑛 ∈ 𝑡) → 𝑥 ∈ 𝑢)
59		elin 3758	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (𝑛 ∩ 𝑢) ↔ (𝑥 ∈ 𝑛 ∧ 𝑥 ∈ 𝑢))
60	59	simplbi2com 655	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑢 → (𝑥 ∈ 𝑛 → 𝑥 ∈ (𝑛 ∩ 𝑢)))
61	58, 60	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)))
62	22	biantrud 527	. . . . . . . . . . . . . . . . . . 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)))
63		ssin 3797	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ⊆ 𝑧 ∧ 𝑛 ⊆ 𝑢) ↔ 𝑛 ⊆ (𝑧 ∩ 𝑢))
64	62, 63	syl6bb 275	. . . . . . . . . . . . . . . . . 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)))
65		ssinss1 3803	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ⊆ 𝑧 → (𝑛 ∩ 𝑢) ⊆ 𝑧)
66	64, 65	syl6bir 243	. . . . . . . . . . . . . . . . 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))
67	61, 66	anim12d 584	. . . . . . . . . . . . . . . 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)))
68	67	reximdva 3000	. . . . . . . . . . . . . . 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)))
69		vex 3176	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑛 ∈ V
70	69	inex1 4727	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∩ 𝑢) ∈ V
71	70	rgenw 2908	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑛 ∈ 𝑡 (𝑛 ∩ 𝑢) ∈ V
72		eleq2 2677	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑛 ∩ 𝑢) → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ (𝑛 ∩ 𝑢)))
73		sseq1 3589	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝑛 ∩ 𝑢) → (𝑤 ⊆ 𝑧 ↔ (𝑛 ∩ 𝑢) ⊆ 𝑧))
74	72, 73	anbi12d 743	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑛 ∩ 𝑢) → ((𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝑥 ∈ (𝑛 ∩ 𝑢) ∧ (𝑛 ∩ 𝑢) ⊆ 𝑧)))
75	29, 74	rexrnmpt 6277	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ 𝑡 (𝑛 ∩ 𝑢) ∈ V → (∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑛 ∈ 𝑡 (𝑥 ∈ (𝑛 ∩ 𝑢) ∧ (𝑛 ∩ 𝑢) ⊆ 𝑧)))
76	71, 75	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑛 ∈ 𝑡 (𝑥 ∈ (𝑛 ∩ 𝑢) ∧ (𝑛 ∩ 𝑢) ⊆ 𝑧))
77	68, 76	syl6ibr 241	. . . . . . . . . . . . . 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)))
78	77	adantrr 749	. . . . . . . . . . . . 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)))
79	78	adantr 480	. . . . . . . . . . . 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)))
80	57, 79	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))
81	80	expr 641	. . . . . . . . . 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)))
82	81	ralrimiva 2949	. . . . . . . . 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)))
83		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑦 = ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) → (𝑦 ≼ ω ↔ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) ≼ ω))
84		rexeq 3116	. . . . . . . . . . . . 13 ⊢ (𝑦 = ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢)) → (∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ ran (𝑎 ∈ 𝑡 ↦ (𝑎 ∩ 𝑢))(𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))
85	84	imbi2d 329	. . . . . . . . . . . 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))))
86	85	ralbidv 2969	. . . . . . . . . . 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))))
87	83, 86	anbi12d 743	. . . . . . . . . 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)))))
88	87	rspcev 3282	. . . . . . . . 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))))
89	36, 39, 82, 88	syl12anc 1316	. . . . . . . 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))))
90	13, 89	rexlimddv 3017	. . . . . . 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))))
91	90	3adantr1 1213	. . . . . 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))))
92		simpl 472	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) → 𝑗 ∈ Locally 1st𝜔)
93	1	adantr 480	. . . . . . . 8 ⊢ ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) → 𝑗 ∈ Top)
94	7	topopn 20536	. . . . . . . 8 ⊢ (𝑗 ∈ Top → ∪ 𝑗 ∈ 𝑗)
95	93, 94	syl 17	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) → ∪ 𝑗 ∈ 𝑗)
96		simpr 476	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) → 𝑥 ∈ ∪ 𝑗)
97		llyi 21087	. . . . . . 7 ⊢ ((𝑗 ∈ Locally 1st𝜔 ∧ ∪ 𝑗 ∈ 𝑗 ∧ 𝑥 ∈ ∪ 𝑗) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ ∪ 𝑗 ∧ 𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔))
98	92, 95, 96, 97	syl3anc 1318	. . . . . 6 ⊢ ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) → ∃𝑢 ∈ 𝑗 (𝑢 ⊆ ∪ 𝑗 ∧ 𝑥 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 1st𝜔))
99	91, 98	r19.29a 3060	. . . . 5 ⊢ ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 ∈ ∪ 𝑗) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
100	99	ralrimiva 2949	. . . 4 ⊢ (𝑗 ∈ Locally 1st𝜔 → ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))
101	7	is1stc2 21055	. . . 4 ⊢ (𝑗 ∈ 1st𝜔 ↔ (𝑗 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → ∃𝑤 ∈ 𝑦 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))))
102	1, 100, 101	sylanbrc 695	. . 3 ⊢ (𝑗 ∈ Locally 1st𝜔 → 𝑗 ∈ 1st𝜔)
103	102	ssriv 3572	. 2 ⊢ Locally 1st𝜔 ⊆ 1st𝜔
104		1stcrest 21066	. . . . 5 ⊢ ((𝑗 ∈ 1st𝜔 ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ 1st𝜔)
105	104	adantl 481	. . . 4 ⊢ ((⊤ ∧ (𝑗 ∈ 1st𝜔 ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ 1st𝜔)
106		1stctop 21056	. . . . . 6 ⊢ (𝑗 ∈ 1st𝜔 → 𝑗 ∈ Top)
107	106	ssriv 3572	. . . . 5 ⊢ 1st𝜔 ⊆ Top
108	107	a1i 11	. . . 4 ⊢ (⊤ → 1st𝜔 ⊆ Top)
109	105, 108	restlly 21096	. . 3 ⊢ (⊤ → 1st𝜔 ⊆ Locally 1st𝜔)
110	109	trud 1484	. 2 ⊢ 1st𝜔 ⊆ Locally 1st𝜔
111	103, 110	eqssi 3584	1 ⊢ Locally 1st𝜔 = 1st𝜔

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  (class class class)co 6549  ωcom 6957   ≼ cdom 7839   ↾_t crest 15904  Topctop 20517  1^st𝜔c1stc 21050  Locally clly 21077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-card 8648  df-acn 8651  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-1stc 21052  df-lly 21079

This theorem is referenced by:  dis1stc  21112