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Theorem lly1stc 22393
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.
StepHypRef Expression
1 llytop 22369 . . . 4 (𝑗 ∈ Locally 1stω → 𝑗 ∈ Top)
2 simprr 773 . . . . . . . . 9 ((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) → (𝑗t 𝑢) ∈ 1stω)
3 simprl 771 . . . . . . . . . 10 ((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) → 𝑥𝑢)
41ad3antrrr 730 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) → 𝑗 ∈ Top)
5 elssuni 4851 . . . . . . . . . . . 12 (𝑢𝑗𝑢 𝑗)
65ad2antlr 727 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) → 𝑢 𝑗)
7 eqid 2737 . . . . . . . . . . . 12 𝑗 = 𝑗
87restuni 22059 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢 𝑗) → 𝑢 = (𝑗t 𝑢))
94, 6, 8syl2anc 587 . . . . . . . . . 10 ((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) → 𝑢 = (𝑗t 𝑢))
103, 9eleqtrd 2840 . . . . . . . . 9 ((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) → 𝑥 (𝑗t 𝑢))
11 eqid 2737 . . . . . . . . . 10 (𝑗t 𝑢) = (𝑗t 𝑢)
12111stcclb 22341 . . . . . . . . 9 (((𝑗t 𝑢) ∈ 1stω ∧ 𝑥 (𝑗t 𝑢)) → ∃𝑡 ∈ 𝒫 (𝑗t 𝑢)(𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))
132, 10, 12syl2anc 587 . . . . . . . 8 ((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) → ∃𝑡 ∈ 𝒫 (𝑗t 𝑢)(𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))
14 elpwi 4522 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ 𝒫 (𝑗t 𝑢) → 𝑡 ⊆ (𝑗t 𝑢))
1514adantl 485 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → 𝑡 ⊆ (𝑗t 𝑢))
1615sselda 3901 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛 ∈ (𝑗t 𝑢))
174adantr 484 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → 𝑗 ∈ Top)
18 simpllr 776 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → 𝑢𝑗)
19 restopn2 22074 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑛 ∈ (𝑗t 𝑢) ↔ (𝑛𝑗𝑛𝑢)))
2017, 18, 19syl2anc 587 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (𝑛 ∈ (𝑗t 𝑢) ↔ (𝑛𝑗𝑛𝑢)))
2120simplbda 503 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛 ∈ (𝑗t 𝑢)) → 𝑛𝑢)
2216, 21syldan 594 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛𝑢)
23 df-ss 3883 . . . . . . . . . . . . . . 15 (𝑛𝑢 ↔ (𝑛𝑢) = 𝑛)
2422, 23sylib 221 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑢) = 𝑛)
2520simprbda 502 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛 ∈ (𝑗t 𝑢)) → 𝑛𝑗)
2616, 25syldan 594 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛𝑗)
2724, 26eqeltrd 2838 . . . . . . . . . . . . 13 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑢) ∈ 𝑗)
28 ineq1 4120 . . . . . . . . . . . . . 14 (𝑎 = 𝑛 → (𝑎𝑢) = (𝑛𝑢))
2928cbvmptv 5158 . . . . . . . . . . . . 13 (𝑎𝑡 ↦ (𝑎𝑢)) = (𝑛𝑡 ↦ (𝑛𝑢))
3027, 29fmptd 6931 . . . . . . . . . . . 12 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (𝑎𝑡 ↦ (𝑎𝑢)):𝑡𝑗)
3130frnd 6553 . . . . . . . . . . 11 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → ran (𝑎𝑡 ↦ (𝑎𝑢)) ⊆ 𝑗)
3231adantrr 717 . . . . . . . . . 10 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → ran (𝑎𝑡 ↦ (𝑎𝑢)) ⊆ 𝑗)
33 vex 3412 . . . . . . . . . . 11 𝑗 ∈ V
3433elpw2 5238 . . . . . . . . . 10 (ran (𝑎𝑡 ↦ (𝑎𝑢)) ∈ 𝒫 𝑗 ↔ ran (𝑎𝑡 ↦ (𝑎𝑢)) ⊆ 𝑗)
3532, 34sylibr 237 . . . . . . . . 9 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → ran (𝑎𝑡 ↦ (𝑎𝑢)) ∈ 𝒫 𝑗)
36 simprrl 781 . . . . . . . . . 10 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → 𝑡 ≼ ω)
37 1stcrestlem 22349 . . . . . . . . . 10 (𝑡 ≼ ω → ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω)
3836, 37syl 17 . . . . . . . . 9 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω)
39 simprr 773 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → 𝑥𝑧)
403ad2antrr 726 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → 𝑥𝑢)
4139, 40elind 4108 . . . . . . . . . . . . 13 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → 𝑥 ∈ (𝑧𝑢))
42 eleq2 2826 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧𝑢) → (𝑥𝑣𝑥 ∈ (𝑧𝑢)))
43 sseq2 3927 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑧𝑢) → (𝑛𝑣𝑛 ⊆ (𝑧𝑢)))
4443anbi2d 632 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑧𝑢) → ((𝑥𝑛𝑛𝑣) ↔ (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
4544rexbidv 3216 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧𝑢) → (∃𝑛𝑡 (𝑥𝑛𝑛𝑣) ↔ ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
4642, 45imbi12d 348 . . . . . . . . . . . . . 14 (𝑣 = (𝑧𝑢) → ((𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣)) ↔ (𝑥 ∈ (𝑧𝑢) → ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)))))
47 simprrr 782 . . . . . . . . . . . . . . 15 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣)))
4847adantr 484 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣)))
494ad2antrr 726 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → 𝑗 ∈ Top)
50 simpllr 776 . . . . . . . . . . . . . . . 16 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → 𝑢𝑗)
5150adantr 484 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → 𝑢𝑗)
52 simprl 771 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → 𝑧𝑗)
53 elrestr 16933 . . . . . . . . . . . . . . 15 ((𝑗 ∈ Top ∧ 𝑢𝑗𝑧𝑗) → (𝑧𝑢) ∈ (𝑗t 𝑢))
5449, 51, 52, 53syl3anc 1373 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → (𝑧𝑢) ∈ (𝑗t 𝑢))
5546, 48, 54rspcdva 3539 . . . . . . . . . . . . 13 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → (𝑥 ∈ (𝑧𝑢) → ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
5641, 55mpd 15 . . . . . . . . . . . 12 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)))
573ad2antrr 726 . . . . . . . . . . . . . . . . . 18 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑥𝑢)
58 elin 3882 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑛𝑢) ↔ (𝑥𝑛𝑥𝑢))
5958simplbi2com 506 . . . . . . . . . . . . . . . . . 18 (𝑥𝑢 → (𝑥𝑛𝑥 ∈ (𝑛𝑢)))
6057, 59syl 17 . . . . . . . . . . . . . . . . 17 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑥𝑛𝑥 ∈ (𝑛𝑢)))
6122biantrud 535 . . . . . . . . . . . . . . . . . . 19 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑧 ↔ (𝑛𝑧𝑛𝑢)))
62 ssin 4145 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑧𝑛𝑢) ↔ 𝑛 ⊆ (𝑧𝑢))
6361, 62bitrdi 290 . . . . . . . . . . . . . . . . . 18 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑧𝑛 ⊆ (𝑧𝑢)))
64 ssinss1 4152 . . . . . . . . . . . . . . . . . 18 (𝑛𝑧 → (𝑛𝑢) ⊆ 𝑧)
6563, 64syl6bir 257 . . . . . . . . . . . . . . . . 17 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛 ⊆ (𝑧𝑢) → (𝑛𝑢) ⊆ 𝑧))
6660, 65anim12d 612 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → ((𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
6766reximdva 3193 . . . . . . . . . . . . . . 15 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → ∃𝑛𝑡 (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
68 vex 3412 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ V
6968inex1 5210 . . . . . . . . . . . . . . . . 17 (𝑛𝑢) ∈ V
7069rgenw 3073 . . . . . . . . . . . . . . . 16 𝑛𝑡 (𝑛𝑢) ∈ V
71 eleq2 2826 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑛𝑢) → (𝑥𝑤𝑥 ∈ (𝑛𝑢)))
72 sseq1 3926 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑛𝑢) → (𝑤𝑧 ↔ (𝑛𝑢) ⊆ 𝑧))
7371, 72anbi12d 634 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑛𝑢) → ((𝑥𝑤𝑤𝑧) ↔ (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
7429, 73rexrnmptw 6914 . . . . . . . . . . . . . . . 16 (∀𝑛𝑡 (𝑛𝑢) ∈ V → (∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧) ↔ ∃𝑛𝑡 (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
7570, 74ax-mp 5 . . . . . . . . . . . . . . 15 (∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧) ↔ ∃𝑛𝑡 (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧))
7667, 75syl6ibr 255 . . . . . . . . . . . . . 14 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
7776adantrr 717 . . . . . . . . . . . . 13 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → (∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
7877adantr 484 . . . . . . . . . . . 12 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → (∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
7956, 78mpd 15 . . . . . . . . . . 11 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧))
8079expr 460 . . . . . . . . . 10 ((((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ 𝑧𝑗) → (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
8180ralrimiva 3105 . . . . . . . . 9 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
82 breq1 5056 . . . . . . . . . . 11 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (𝑦 ≼ ω ↔ ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω))
83 rexeq 3320 . . . . . . . . . . . . 13 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
8483imbi2d 344 . . . . . . . . . . . 12 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧))))
8584ralbidv 3118 . . . . . . . . . . 11 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧))))
8682, 85anbi12d 634 . . . . . . . . . 10 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → ((𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ (ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))))
8786rspcev 3537 . . . . . . . . 9 ((ran (𝑎𝑡 ↦ (𝑎𝑢)) ∈ 𝒫 𝑗 ∧ (ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
8835, 38, 81, 87syl12anc 837 . . . . . . . 8 (((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
8913, 88rexlimddv 3210 . . . . . . 7 ((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
90893adantr1 1171 . . . . . 6 ((((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
91 simpl 486 . . . . . . 7 ((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) → 𝑗 ∈ Locally 1stω)
921adantr 484 . . . . . . . 8 ((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) → 𝑗 ∈ Top)
937topopn 21803 . . . . . . . 8 (𝑗 ∈ Top → 𝑗𝑗)
9492, 93syl 17 . . . . . . 7 ((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) → 𝑗𝑗)
95 simpr 488 . . . . . . 7 ((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) → 𝑥 𝑗)
96 llyi 22371 . . . . . . 7 ((𝑗 ∈ Locally 1stω ∧ 𝑗𝑗𝑥 𝑗) → ∃𝑢𝑗 (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω))
9791, 94, 95, 96syl3anc 1373 . . . . . 6 ((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) → ∃𝑢𝑗 (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1stω))
9890, 97r19.29a 3208 . . . . 5 ((𝑗 ∈ Locally 1stω ∧ 𝑥 𝑗) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
9998ralrimiva 3105 . . . 4 (𝑗 ∈ Locally 1stω → ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1007is1stc2 22339 . . . 4 (𝑗 ∈ 1stω ↔ (𝑗 ∈ Top ∧ ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
1011, 99, 100sylanbrc 586 . . 3 (𝑗 ∈ Locally 1stω → 𝑗 ∈ 1stω)
102101ssriv 3905 . 2 Locally 1stω ⊆ 1stω
103 1stcrest 22350 . . . . 5 ((𝑗 ∈ 1stω ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ 1stω)
104103adantl 485 . . . 4 ((⊤ ∧ (𝑗 ∈ 1stω ∧ 𝑥𝑗)) → (𝑗t 𝑥) ∈ 1stω)
105 1stctop 22340 . . . . . 6 (𝑗 ∈ 1stω → 𝑗 ∈ Top)
106105ssriv 3905 . . . . 5 1stω ⊆ Top
107106a1i 11 . . . 4 (⊤ → 1stω ⊆ Top)
108104, 107restlly 22380 . . 3 (⊤ → 1stω ⊆ Locally 1stω)
109108mptru 1550 . 2 1stω ⊆ Locally 1stω
110102, 109eqssi 3917 1 Locally 1stω = 1stω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wtru 1544  wcel 2110  wral 3061  wrex 3062  Vcvv 3408  cin 3865  wss 3866  𝒫 cpw 4513   cuni 4819   class class class wbr 5053  cmpt 5135  ran crn 5552  (class class class)co 7213  ωcom 7644  cdom 8624  t crest 16925  Topctop 21790  1stωc1stc 22334  Locally clly 22361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-fin 8630  df-fi 9027  df-card 9555  df-acn 9558  df-rest 16927  df-topgen 16948  df-top 21791  df-topon 21808  df-bases 21843  df-1stc 22336  df-lly 22363
This theorem is referenced by:  dis1stc  22396
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