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Theorem lly1stc 21209
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 21185 . . . 4 (𝑗 ∈ Locally 1st𝜔 → 𝑗 ∈ Top)
2 simprr 795 . . . . . . . . 9 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → (𝑗t 𝑢) ∈ 1st𝜔)
3 simprl 793 . . . . . . . . . 10 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑥𝑢)
41ad3antrrr 765 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑗 ∈ Top)
5 elssuni 4433 . . . . . . . . . . . 12 (𝑢𝑗𝑢 𝑗)
65ad2antlr 762 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑢 𝑗)
7 eqid 2621 . . . . . . . . . . . 12 𝑗 = 𝑗
87restuni 20876 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢 𝑗) → 𝑢 = (𝑗t 𝑢))
94, 6, 8syl2anc 692 . . . . . . . . . 10 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑢 = (𝑗t 𝑢))
103, 9eleqtrd 2700 . . . . . . . . 9 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑥 (𝑗t 𝑢))
11 eqid 2621 . . . . . . . . . 10 (𝑗t 𝑢) = (𝑗t 𝑢)
12111stcclb 21157 . . . . . . . . 9 (((𝑗t 𝑢) ∈ 1st𝜔 ∧ 𝑥 (𝑗t 𝑢)) → ∃𝑡 ∈ 𝒫 (𝑗t 𝑢)(𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))
132, 10, 12syl2anc 692 . . . . . . . 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 4140 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ 𝒫 (𝑗t 𝑢) → 𝑡 ⊆ (𝑗t 𝑢))
1514adantl 482 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → 𝑡 ⊆ (𝑗t 𝑢))
1615sselda 3583 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛 ∈ (𝑗t 𝑢))
174adantr 481 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → 𝑗 ∈ Top)
18 simpllr 798 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → 𝑢𝑗)
19 restopn2 20891 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑛 ∈ (𝑗t 𝑢) ↔ (𝑛𝑗𝑛𝑢)))
2017, 18, 19syl2anc 692 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (𝑛 ∈ (𝑗t 𝑢) ↔ (𝑛𝑗𝑛𝑢)))
2120simplbda 653 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛 ∈ (𝑗t 𝑢)) → 𝑛𝑢)
2216, 21syldan 487 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛𝑢)
23 df-ss 3569 . . . . . . . . . . . . . . 15 (𝑛𝑢 ↔ (𝑛𝑢) = 𝑛)
2422, 23sylib 208 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑢) = 𝑛)
2520simprbda 652 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛 ∈ (𝑗t 𝑢)) → 𝑛𝑗)
2616, 25syldan 487 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛𝑗)
2724, 26eqeltrd 2698 . . . . . . . . . . . . 13 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑢) ∈ 𝑗)
28 ineq1 3785 . . . . . . . . . . . . . 14 (𝑎 = 𝑛 → (𝑎𝑢) = (𝑛𝑢))
2928cbvmptv 4710 . . . . . . . . . . . . 13 (𝑎𝑡 ↦ (𝑎𝑢)) = (𝑛𝑡 ↦ (𝑛𝑢))
3027, 29fmptd 6340 . . . . . . . . . . . 12 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (𝑎𝑡 ↦ (𝑎𝑢)):𝑡𝑗)
31 frn 6010 . . . . . . . . . . . 12 ((𝑎𝑡 ↦ (𝑎𝑢)):𝑡𝑗 → ran (𝑎𝑡 ↦ (𝑎𝑢)) ⊆ 𝑗)
3230, 31syl 17 . . . . . . . . . . 11 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → ran (𝑎𝑡 ↦ (𝑎𝑢)) ⊆ 𝑗)
3332adantrr 752 . . . . . . . . . 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)
34 vex 3189 . . . . . . . . . . 11 𝑗 ∈ V
3534elpw2 4788 . . . . . . . . . 10 (ran (𝑎𝑡 ↦ (𝑎𝑢)) ∈ 𝒫 𝑗 ↔ ran (𝑎𝑡 ↦ (𝑎𝑢)) ⊆ 𝑗)
3633, 35sylibr 224 . . . . . . . . 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)
37 simprrl 803 . . . . . . . . . 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)
38 1stcrestlem 21165 . . . . . . . . . 10 (𝑡 ≼ ω → ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω)
3937, 38syl 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)
404ad2antrr 761 . . . . . . . . . . . . . 14 ((((((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)
41 simpllr 798 . . . . . . . . . . . . . . 15 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → 𝑢𝑗)
4241adantr 481 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → 𝑢𝑗)
43 simprl 793 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) ∧ (𝑧𝑗𝑥𝑧)) → 𝑧𝑗)
44 elrestr 16010 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑢𝑗𝑧𝑗) → (𝑧𝑢) ∈ (𝑗t 𝑢))
4540, 42, 43, 44syl3anc 1323 . . . . . . . . . . . . 13 ((((((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))
46 simprrr 804 . . . . . . . . . . . . . 14 (((((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)))
4746adantr 481 . . . . . . . . . . . . 13 ((((((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)))
48 simprr 795 . . . . . . . . . . . . . 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)
493ad2antrr 761 . . . . . . . . . . . . . 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)
5048, 49elind 3776 . . . . . . . . . . . . 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))
51 eleq2 2687 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧𝑢) → (𝑥𝑣𝑥 ∈ (𝑧𝑢)))
52 sseq2 3606 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑧𝑢) → (𝑛𝑣𝑛 ⊆ (𝑧𝑢)))
5352anbi2d 739 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑧𝑢) → ((𝑥𝑛𝑛𝑣) ↔ (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
5453rexbidv 3045 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧𝑢) → (∃𝑛𝑡 (𝑥𝑛𝑛𝑣) ↔ ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
5551, 54imbi12d 334 . . . . . . . . . . . . . 14 (𝑣 = (𝑧𝑢) → ((𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣)) ↔ (𝑥 ∈ (𝑧𝑢) → ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)))))
5655rspcv 3291 . . . . . . . . . . . . 13 ((𝑧𝑢) ∈ (𝑗t 𝑢) → (∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣)) → (𝑥 ∈ (𝑧𝑢) → ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)))))
5745, 47, 50, 56syl3c 66 . . . . . . . . . . . 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)))
583ad2antrr 761 . . . . . . . . . . . . . . . . . 18 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑥𝑢)
59 elin 3774 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑛𝑢) ↔ (𝑥𝑛𝑥𝑢))
6059simplbi2com 656 . . . . . . . . . . . . . . . . . 18 (𝑥𝑢 → (𝑥𝑛𝑥 ∈ (𝑛𝑢)))
6158, 60syl 17 . . . . . . . . . . . . . . . . 17 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑥𝑛𝑥 ∈ (𝑛𝑢)))
6222biantrud 528 . . . . . . . . . . . . . . . . . . 19 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑧 ↔ (𝑛𝑧𝑛𝑢)))
63 ssin 3813 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑧𝑛𝑢) ↔ 𝑛 ⊆ (𝑧𝑢))
6462, 63syl6bb 276 . . . . . . . . . . . . . . . . . 18 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑧𝑛 ⊆ (𝑧𝑢)))
65 ssinss1 3819 . . . . . . . . . . . . . . . . . 18 (𝑛𝑧 → (𝑛𝑢) ⊆ 𝑧)
6664, 65syl6bir 244 . . . . . . . . . . . . . . . . 17 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛 ⊆ (𝑧𝑢) → (𝑛𝑢) ⊆ 𝑧))
6761, 66anim12d 585 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → ((𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
6867reximdva 3011 . . . . . . . . . . . . . . 15 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → ∃𝑛𝑡 (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
69 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ V
7069inex1 4759 . . . . . . . . . . . . . . . . 17 (𝑛𝑢) ∈ V
7170rgenw 2919 . . . . . . . . . . . . . . . 16 𝑛𝑡 (𝑛𝑢) ∈ V
72 eleq2 2687 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑛𝑢) → (𝑥𝑤𝑥 ∈ (𝑛𝑢)))
73 sseq1 3605 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑛𝑢) → (𝑤𝑧 ↔ (𝑛𝑢) ⊆ 𝑧))
7472, 73anbi12d 746 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑛𝑢) → ((𝑥𝑤𝑤𝑧) ↔ (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
7529, 74rexrnmpt 6325 . . . . . . . . . . . . . . . 16 (∀𝑛𝑡 (𝑛𝑢) ∈ V → (∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧) ↔ ∃𝑛𝑡 (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
7671, 75ax-mp 5 . . . . . . . . . . . . . . 15 (∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧) ↔ ∃𝑛𝑡 (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧))
7768, 76syl6ibr 242 . . . . . . . . . . . . . 14 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
7877adantrr 752 . . . . . . . . . . . . 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)))
7978adantr 481 . . . . . . . . . . . 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)))
8057, 79mpd 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))
8180expr 642 . . . . . . . . . 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)))
8281ralrimiva 2960 . . . . . . . . 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)))
83 breq1 4616 . . . . . . . . . . 11 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (𝑦 ≼ ω ↔ ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω))
84 rexeq 3128 . . . . . . . . . . . . 13 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
8584imbi2d 330 . . . . . . . . . . . 12 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧))))
8685ralbidv 2980 . . . . . . . . . . 11 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧))))
8783, 86anbi12d 746 . . . . . . . . . 10 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → ((𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ (ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))))
8887rspcev 3295 . . . . . . . . 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))))
8936, 39, 82, 88syl12anc 1321 . . . . . . . 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))))
9013, 89rexlimddv 3028 . . . . . . 7 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
91903adantr1 1218 . . . . . 6 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
92 simpl 473 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑗 ∈ Locally 1st𝜔)
931adantr 481 . . . . . . . 8 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑗 ∈ Top)
947topopn 20636 . . . . . . . 8 (𝑗 ∈ Top → 𝑗𝑗)
9593, 94syl 17 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑗𝑗)
96 simpr 477 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑥 𝑗)
97 llyi 21187 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑗𝑗𝑥 𝑗) → ∃𝑢𝑗 (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔))
9892, 95, 96, 97syl3anc 1323 . . . . . 6 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → ∃𝑢𝑗 (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔))
9991, 98r19.29a 3071 . . . . 5 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
10099ralrimiva 2960 . . . 4 (𝑗 ∈ Locally 1st𝜔 → ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1017is1stc2 21155 . . . 4 (𝑗 ∈ 1st𝜔 ↔ (𝑗 ∈ Top ∧ ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
1021, 100, 101sylanbrc 697 . . 3 (𝑗 ∈ Locally 1st𝜔 → 𝑗 ∈ 1st𝜔)
103102ssriv 3587 . 2 Locally 1st𝜔 ⊆ 1st𝜔
104 1stcrest 21166 . . . . 5 ((𝑗 ∈ 1st𝜔 ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ 1st𝜔)
105104adantl 482 . . . 4 ((⊤ ∧ (𝑗 ∈ 1st𝜔 ∧ 𝑥𝑗)) → (𝑗t 𝑥) ∈ 1st𝜔)
106 1stctop 21156 . . . . . 6 (𝑗 ∈ 1st𝜔 → 𝑗 ∈ Top)
107106ssriv 3587 . . . . 5 1st𝜔 ⊆ Top
108107a1i 11 . . . 4 (⊤ → 1st𝜔 ⊆ Top)
109105, 108restlly 21196 . . 3 (⊤ → 1st𝜔 ⊆ Locally 1st𝜔)
110109trud 1490 . 2 1st𝜔 ⊆ Locally 1st𝜔
111103, 110eqssi 3599 1 Locally 1st𝜔 = 1st𝜔
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wtru 1481  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  cmpt 4673  ran crn 5075  wf 5843  (class class class)co 6604  ωcom 7012  cdom 7897  t crest 16002  Topctop 20617  1st𝜔c1stc 21150  Locally clly 21177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-fin 7903  df-fi 8261  df-card 8709  df-acn 8712  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-1stc 21152  df-lly 21179
This theorem is referenced by:  dis1stc  21212
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