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Theorem lly1stc 21593
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 21569 . . . 4 (𝑗 ∈ Locally 1st𝜔 → 𝑗 ∈ Top)
2 simprr 789 . . . . . . . . 9 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → (𝑗t 𝑢) ∈ 1st𝜔)
3 simprl 787 . . . . . . . . . 10 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑥𝑢)
41ad3antrrr 721 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑗 ∈ Top)
5 elssuni 4627 . . . . . . . . . . . 12 (𝑢𝑗𝑢 𝑗)
65ad2antlr 718 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑢 𝑗)
7 eqid 2765 . . . . . . . . . . . 12 𝑗 = 𝑗
87restuni 21260 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢 𝑗) → 𝑢 = (𝑗t 𝑢))
94, 6, 8syl2anc 579 . . . . . . . . . 10 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑢 = (𝑗t 𝑢))
103, 9eleqtrd 2846 . . . . . . . . 9 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑥 (𝑗t 𝑢))
11 eqid 2765 . . . . . . . . . 10 (𝑗t 𝑢) = (𝑗t 𝑢)
12111stcclb 21541 . . . . . . . . 9 (((𝑗t 𝑢) ∈ 1st𝜔 ∧ 𝑥 (𝑗t 𝑢)) → ∃𝑡 ∈ 𝒫 (𝑗t 𝑢)(𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))
132, 10, 12syl2anc 579 . . . . . . . 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 4327 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ 𝒫 (𝑗t 𝑢) → 𝑡 ⊆ (𝑗t 𝑢))
1514adantl 473 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → 𝑡 ⊆ (𝑗t 𝑢))
1615sselda 3763 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛 ∈ (𝑗t 𝑢))
174adantr 472 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → 𝑗 ∈ Top)
18 simpllr 793 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → 𝑢𝑗)
19 restopn2 21275 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑛 ∈ (𝑗t 𝑢) ↔ (𝑛𝑗𝑛𝑢)))
2017, 18, 19syl2anc 579 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (𝑛 ∈ (𝑗t 𝑢) ↔ (𝑛𝑗𝑛𝑢)))
2120simplbda 493 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛 ∈ (𝑗t 𝑢)) → 𝑛𝑢)
2216, 21syldan 585 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛𝑢)
23 df-ss 3748 . . . . . . . . . . . . . . 15 (𝑛𝑢 ↔ (𝑛𝑢) = 𝑛)
2422, 23sylib 209 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑢) = 𝑛)
2520simprbda 492 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛 ∈ (𝑗t 𝑢)) → 𝑛𝑗)
2616, 25syldan 585 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛𝑗)
2724, 26eqeltrd 2844 . . . . . . . . . . . . 13 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑢) ∈ 𝑗)
28 ineq1 3971 . . . . . . . . . . . . . 14 (𝑎 = 𝑛 → (𝑎𝑢) = (𝑛𝑢))
2928cbvmptv 4911 . . . . . . . . . . . . 13 (𝑎𝑡 ↦ (𝑎𝑢)) = (𝑛𝑡 ↦ (𝑛𝑢))
3027, 29fmptd 6578 . . . . . . . . . . . 12 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (𝑎𝑡 ↦ (𝑎𝑢)):𝑡𝑗)
3130frnd 6232 . . . . . . . . . . 11 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → ran (𝑎𝑡 ↦ (𝑎𝑢)) ⊆ 𝑗)
3231adantrr 708 . . . . . . . . . 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 3353 . . . . . . . . . . 11 𝑗 ∈ V
3433elpw2 4988 . . . . . . . . . 10 (ran (𝑎𝑡 ↦ (𝑎𝑢)) ∈ 𝒫 𝑗 ↔ ran (𝑎𝑡 ↦ (𝑎𝑢)) ⊆ 𝑗)
3532, 34sylibr 225 . . . . . . . . 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 799 . . . . . . . . . 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 21549 . . . . . . . . . 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 789 . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . 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 3962 . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧𝑢) → (𝑥𝑣𝑥 ∈ (𝑧𝑢)))
43 sseq2 3789 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑧𝑢) → (𝑛𝑣𝑛 ⊆ (𝑧𝑢)))
4443anbi2d 622 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑧𝑢) → ((𝑥𝑛𝑛𝑣) ↔ (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
4544rexbidv 3199 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧𝑢) → (∃𝑛𝑡 (𝑥𝑛𝑛𝑣) ↔ ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
4642, 45imbi12d 335 . . . . . . . . . . . . . 14 (𝑣 = (𝑧𝑢) → ((𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣)) ↔ (𝑥 ∈ (𝑧𝑢) → ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)))))
47 simprrr 800 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . 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 793 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . 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 16369 . . . . . . . . . . . . . . 15 ((𝑗 ∈ Top ∧ 𝑢𝑗𝑧𝑗) → (𝑧𝑢) ∈ (𝑗t 𝑢))
5449, 51, 52, 53syl3anc 1490 . . . . . . . . . . . . . 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 3468 . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . . . 18 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑥𝑢)
58 elin 3960 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑛𝑢) ↔ (𝑥𝑛𝑥𝑢))
5958simplbi2com 496 . . . . . . . . . . . . . . . . . 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 527 . . . . . . . . . . . . . . . . . . 19 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑧 ↔ (𝑛𝑧𝑛𝑢)))
62 ssin 3996 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑧𝑛𝑢) ↔ 𝑛 ⊆ (𝑧𝑢))
6361, 62syl6bb 278 . . . . . . . . . . . . . . . . . 18 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑧𝑛 ⊆ (𝑧𝑢)))
64 ssinss1 4003 . . . . . . . . . . . . . . . . . 18 (𝑛𝑧 → (𝑛𝑢) ⊆ 𝑧)
6563, 64syl6bir 245 . . . . . . . . . . . . . . . . 17 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛 ⊆ (𝑧𝑢) → (𝑛𝑢) ⊆ 𝑧))
6660, 65anim12d 602 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → ((𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
6766reximdva 3163 . . . . . . . . . . . . . . 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 3353 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ V
6968inex1 4962 . . . . . . . . . . . . . . . . 17 (𝑛𝑢) ∈ V
7069rgenw 3071 . . . . . . . . . . . . . . . 16 𝑛𝑡 (𝑛𝑢) ∈ V
71 eleq2 2833 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑛𝑢) → (𝑥𝑤𝑥 ∈ (𝑛𝑢)))
72 sseq1 3788 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑛𝑢) → (𝑤𝑧 ↔ (𝑛𝑢) ⊆ 𝑧))
7371, 72anbi12d 624 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑛𝑢) → ((𝑥𝑤𝑤𝑧) ↔ (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
7429, 73rexrnmpt 6563 . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . 14 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
7776adantrr 708 . . . . . . . . . . . . 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 472 . . . . . . . . . . . 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 448 . . . . . . . . . 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 3113 . . . . . . . . 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 4814 . . . . . . . . . . 11 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (𝑦 ≼ ω ↔ ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω))
83 rexeq 3287 . . . . . . . . . . . . 13 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
8483imbi2d 331 . . . . . . . . . . . 12 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧))))
8584ralbidv 3133 . . . . . . . . . . 11 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧))))
8682, 85anbi12d 624 . . . . . . . . . 10 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → ((𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ (ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))))
8786rspcev 3462 . . . . . . . . 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 865 . . . . . . . 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 3182 . . . . . . 7 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
90893adantr1 1210 . . . . . 6 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
91 simpl 474 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑗 ∈ Locally 1st𝜔)
921adantr 472 . . . . . . . 8 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑗 ∈ Top)
937topopn 21004 . . . . . . . 8 (𝑗 ∈ Top → 𝑗𝑗)
9492, 93syl 17 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑗𝑗)
95 simpr 477 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑥 𝑗)
96 llyi 21571 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑗𝑗𝑥 𝑗) → ∃𝑢𝑗 (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔))
9791, 94, 95, 96syl3anc 1490 . . . . . 6 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → ∃𝑢𝑗 (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔))
9890, 97r19.29a 3225 . . . . 5 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
9998ralrimiva 3113 . . . 4 (𝑗 ∈ Locally 1st𝜔 → ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1007is1stc2 21539 . . . 4 (𝑗 ∈ 1st𝜔 ↔ (𝑗 ∈ Top ∧ ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
1011, 99, 100sylanbrc 578 . . 3 (𝑗 ∈ Locally 1st𝜔 → 𝑗 ∈ 1st𝜔)
102101ssriv 3767 . 2 Locally 1st𝜔 ⊆ 1st𝜔
103 1stcrest 21550 . . . . 5 ((𝑗 ∈ 1st𝜔 ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ 1st𝜔)
104103adantl 473 . . . 4 ((⊤ ∧ (𝑗 ∈ 1st𝜔 ∧ 𝑥𝑗)) → (𝑗t 𝑥) ∈ 1st𝜔)
105 1stctop 21540 . . . . . 6 (𝑗 ∈ 1st𝜔 → 𝑗 ∈ Top)
106105ssriv 3767 . . . . 5 1st𝜔 ⊆ Top
107106a1i 11 . . . 4 (⊤ → 1st𝜔 ⊆ Top)
108104, 107restlly 21580 . . 3 (⊤ → 1st𝜔 ⊆ Locally 1st𝜔)
109108mptru 1660 . 2 1st𝜔 ⊆ Locally 1st𝜔
110102, 109eqssi 3779 1 Locally 1st𝜔 = 1st𝜔
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wtru 1653  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cin 3733  wss 3734  𝒫 cpw 4317   cuni 4596   class class class wbr 4811  cmpt 4890  ran crn 5280  (class class class)co 6846  ωcom 7267  cdom 8162  t crest 16361  Topctop 20991  1st𝜔c1stc 21534  Locally clly 21561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-oadd 7772  df-er 7951  df-map 8066  df-en 8165  df-dom 8166  df-fin 8168  df-fi 8528  df-card 9020  df-acn 9023  df-rest 16363  df-topgen 16384  df-top 20992  df-topon 21009  df-bases 21044  df-1stc 21536  df-lly 21563
This theorem is referenced by:  dis1stc  21596
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