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Theorem lly1stc 21501
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 21477 . . . 4 (𝑗 ∈ Locally 1st𝜔 → 𝑗 ∈ Top)
2 simprr 813 . . . . . . . . 9 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → (𝑗t 𝑢) ∈ 1st𝜔)
3 simprl 811 . . . . . . . . . 10 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑥𝑢)
41ad3antrrr 768 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑗 ∈ Top)
5 elssuni 4619 . . . . . . . . . . . 12 (𝑢𝑗𝑢 𝑗)
65ad2antlr 765 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑢 𝑗)
7 eqid 2760 . . . . . . . . . . . 12 𝑗 = 𝑗
87restuni 21168 . . . . . . . . . . 11 ((𝑗 ∈ Top ∧ 𝑢 𝑗) → 𝑢 = (𝑗t 𝑢))
94, 6, 8syl2anc 696 . . . . . . . . . 10 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑢 = (𝑗t 𝑢))
103, 9eleqtrd 2841 . . . . . . . . 9 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑥 (𝑗t 𝑢))
11 eqid 2760 . . . . . . . . . 10 (𝑗t 𝑢) = (𝑗t 𝑢)
12111stcclb 21449 . . . . . . . . 9 (((𝑗t 𝑢) ∈ 1st𝜔 ∧ 𝑥 (𝑗t 𝑢)) → ∃𝑡 ∈ 𝒫 (𝑗t 𝑢)(𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))
132, 10, 12syl2anc 696 . . . . . . . 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 4312 . . . . . . . . . . . . . . . . . 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 3744 . . . . . . . . . . . . . . . 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 817 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → 𝑢𝑗)
19 restopn2 21183 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑢𝑗) → (𝑛 ∈ (𝑗t 𝑢) ↔ (𝑛𝑗𝑛𝑢)))
2017, 18, 19syl2anc 696 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (𝑛 ∈ (𝑗t 𝑢) ↔ (𝑛𝑗𝑛𝑢)))
2120simplbda 655 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛 ∈ (𝑗t 𝑢)) → 𝑛𝑢)
2216, 21syldan 488 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛𝑢)
23 df-ss 3729 . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . 15 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛 ∈ (𝑗t 𝑢)) → 𝑛𝑗)
2616, 25syldan 488 . . . . . . . . . . . . . 14 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑛𝑗)
2724, 26eqeltrd 2839 . . . . . . . . . . . . 13 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑢) ∈ 𝑗)
28 ineq1 3950 . . . . . . . . . . . . . 14 (𝑎 = 𝑛 → (𝑎𝑢) = (𝑛𝑢))
2928cbvmptv 4902 . . . . . . . . . . . . 13 (𝑎𝑡 ↦ (𝑎𝑢)) = (𝑛𝑡 ↦ (𝑛𝑢))
3027, 29fmptd 6548 . . . . . . . . . . . 12 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (𝑎𝑡 ↦ (𝑎𝑢)):𝑡𝑗)
31 frn 6214 . . . . . . . . . . . 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 755 . . . . . . . . . 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 3343 . . . . . . . . . . 11 𝑗 ∈ V
3534elpw2 4977 . . . . . . . . . 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 823 . . . . . . . . . 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 21457 . . . . . . . . . 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)
40 simprr 813 . . . . . . . . . . . . . 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)
413ad2antrr 764 . . . . . . . . . . . . . 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)
4240, 41elind 3941 . . . . . . . . . . . . 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))
43 eleq2 2828 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧𝑢) → (𝑥𝑣𝑥 ∈ (𝑧𝑢)))
44 sseq2 3768 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑧𝑢) → (𝑛𝑣𝑛 ⊆ (𝑧𝑢)))
4544anbi2d 742 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑧𝑢) → ((𝑥𝑛𝑛𝑣) ↔ (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
4645rexbidv 3190 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧𝑢) → (∃𝑛𝑡 (𝑥𝑛𝑛𝑣) ↔ ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
4743, 46imbi12d 333 . . . . . . . . . . . . . 14 (𝑣 = (𝑧𝑢) → ((𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣)) ↔ (𝑥 ∈ (𝑧𝑢) → ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)))))
48 simprrr 824 . . . . . . . . . . . . . . 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)))
4948adantr 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)))
504ad2antrr 764 . . . . . . . . . . . . . . 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)
51 simpllr 817 . . . . . . . . . . . . . . . 16 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ (𝑡 ∈ 𝒫 (𝑗t 𝑢) ∧ (𝑡 ≼ ω ∧ ∀𝑣 ∈ (𝑗t 𝑢)(𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣))))) → 𝑢𝑗)
5251adantr 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)
53 simprl 811 . . . . . . . . . . . . . . 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)
54 elrestr 16291 . . . . . . . . . . . . . . 15 ((𝑗 ∈ Top ∧ 𝑢𝑗𝑧𝑗) → (𝑧𝑢) ∈ (𝑗t 𝑢))
5550, 52, 53, 54syl3anc 1477 . . . . . . . . . . . . . 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))
5647, 49, 55rspcdva 3455 . . . . . . . . . . . . 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))))
5742, 56mpd 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)))
583ad2antrr 764 . . . . . . . . . . . . . . . . . 18 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑥𝑢)
59 elin 3939 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝑛𝑢) ↔ (𝑥𝑛𝑥𝑢))
6059simplbi2com 658 . . . . . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . . . . . . 19 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑧 ↔ (𝑛𝑧𝑛𝑢)))
63 ssin 3978 . . . . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . 16 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → ((𝑥𝑛𝑛 ⊆ (𝑧𝑢)) → (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
6867reximdva 3155 . . . . . . . . . . . . . . 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 3343 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ V
7069inex1 4951 . . . . . . . . . . . . . . . . 17 (𝑛𝑢) ∈ V
7170rgenw 3062 . . . . . . . . . . . . . . . 16 𝑛𝑡 (𝑛𝑢) ∈ V
72 eleq2 2828 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑛𝑢) → (𝑥𝑤𝑥 ∈ (𝑛𝑢)))
73 sseq1 3767 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑛𝑢) → (𝑤𝑧 ↔ (𝑛𝑢) ⊆ 𝑧))
7472, 73anbi12d 749 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑛𝑢) → ((𝑥𝑤𝑤𝑧) ↔ (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
7529, 74rexrnmpt 6532 . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . 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 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)))
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 644 . . . . . . . . . 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 3104 . . . . . . . . 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 4807 . . . . . . . . . . 11 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (𝑦 ≼ ω ↔ ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω))
84 rexeq 3278 . . . . . . . . . . . . 13 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (∃𝑤𝑦 (𝑥𝑤𝑤𝑧) ↔ ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))
8584imbi2d 329 . . . . . . . . . . . 12 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → ((𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧))))
8685ralbidv 3124 . . . . . . . . . . 11 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)) ↔ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧))))
8783, 86anbi12d 749 . . . . . . . . . 10 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → ((𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))) ↔ (ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤 ∈ ran (𝑎𝑡 ↦ (𝑎𝑢))(𝑥𝑤𝑤𝑧)))))
8887rspcev 3449 . . . . . . . . 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 1475 . . . . . . . 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 3173 . . . . . . 7 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
91903adantr1 1175 . . . . . 6 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
92 simpl 474 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑗 ∈ Locally 1st𝜔)
931adantr 472 . . . . . . . 8 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑗 ∈ Top)
947topopn 20913 . . . . . . . 8 (𝑗 ∈ Top → 𝑗𝑗)
9593, 94syl 17 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑗𝑗)
96 simpr 479 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → 𝑥 𝑗)
97 llyi 21479 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑗𝑗𝑥 𝑗) → ∃𝑢𝑗 (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔))
9892, 95, 96, 97syl3anc 1477 . . . . . 6 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → ∃𝑢𝑗 (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔))
9991, 98r19.29a 3216 . . . . 5 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
10099ralrimiva 3104 . . . 4 (𝑗 ∈ Locally 1st𝜔 → ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1017is1stc2 21447 . . . 4 (𝑗 ∈ 1st𝜔 ↔ (𝑗 ∈ Top ∧ ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧)))))
1021, 100, 101sylanbrc 701 . . 3 (𝑗 ∈ Locally 1st𝜔 → 𝑗 ∈ 1st𝜔)
103102ssriv 3748 . 2 Locally 1st𝜔 ⊆ 1st𝜔
104 1stcrest 21458 . . . . 5 ((𝑗 ∈ 1st𝜔 ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ 1st𝜔)
105104adantl 473 . . . 4 ((⊤ ∧ (𝑗 ∈ 1st𝜔 ∧ 𝑥𝑗)) → (𝑗t 𝑥) ∈ 1st𝜔)
106 1stctop 21448 . . . . . 6 (𝑗 ∈ 1st𝜔 → 𝑗 ∈ Top)
107106ssriv 3748 . . . . 5 1st𝜔 ⊆ Top
108107a1i 11 . . . 4 (⊤ → 1st𝜔 ⊆ Top)
109105, 108restlly 21488 . . 3 (⊤ → 1st𝜔 ⊆ Locally 1st𝜔)
110109trud 1642 . 2 1st𝜔 ⊆ Locally 1st𝜔
111103, 110eqssi 3760 1 Locally 1st𝜔 = 1st𝜔
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wtru 1633  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588   class class class wbr 4804  cmpt 4881  ran crn 5267  wf 6045  (class class class)co 6813  ωcom 7230  cdom 8119  t crest 16283  Topctop 20900  1st𝜔c1stc 21442  Locally clly 21469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-fin 8125  df-fi 8482  df-card 8955  df-acn 8958  df-rest 16285  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-1stc 21444  df-lly 21471
This theorem is referenced by:  dis1stc  21504
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