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Theorem lly1stc 21493
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 21469 . . . 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 4611 . . . . . . . . . . . 12 (𝑢𝑗𝑢 𝑗)
65ad2antlr 765 . . . . . . . . . . 11 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑢 𝑗)
7 eqid 2752 . . . . . . . . . . . 12 𝑗 = 𝑗
87restuni 21160 . . . . . . . . . . 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 2833 . . . . . . . . 9 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → 𝑥 (𝑗t 𝑢))
11 eqid 2752 . . . . . . . . . 10 (𝑗t 𝑢) = (𝑗t 𝑢)
12111stcclb 21441 . . . . . . . . 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 4304 . . . . . . . . . . . . . . . . . 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 3736 . . . . . . . . . . . . . . . 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 21175 . . . . . . . . . . . . . . . . . 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 3721 . . . . . . . . . . . . . . 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 2831 . . . . . . . . . . . . 13 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → (𝑛𝑢) ∈ 𝑗)
28 ineq1 3942 . . . . . . . . . . . . . 14 (𝑎 = 𝑛 → (𝑎𝑢) = (𝑛𝑢))
2928cbvmptv 4894 . . . . . . . . . . . . 13 (𝑎𝑡 ↦ (𝑎𝑢)) = (𝑛𝑡 ↦ (𝑛𝑢))
3027, 29fmptd 6540 . . . . . . . . . . . 12 (((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) → (𝑎𝑡 ↦ (𝑎𝑢)):𝑡𝑗)
31 frn 6206 . . . . . . . . . . . 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 3335 . . . . . . . . . . 11 𝑗 ∈ V
3534elpw2 4969 . . . . . . . . . 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 21449 . . . . . . . . . 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 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)) → 𝑗 ∈ Top)
41 simpllr 817 . . . . . . . . . . . . . . 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 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)) → 𝑢𝑗)
43 simprl 811 . . . . . . . . . . . . . 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 16283 . . . . . . . . . . . . . 14 ((𝑗 ∈ Top ∧ 𝑢𝑗𝑧𝑗) → (𝑧𝑢) ∈ (𝑗t 𝑢))
4540, 42, 43, 44syl3anc 1473 . . . . . . . . . . . . 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 824 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 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)
493ad2antrr 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)
5048, 49elind 3933 . . . . . . . . . . . . 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 2820 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧𝑢) → (𝑥𝑣𝑥 ∈ (𝑧𝑢)))
52 sseq2 3760 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑧𝑢) → (𝑛𝑣𝑛 ⊆ (𝑧𝑢)))
5352anbi2d 742 . . . . . . . . . . . . . . . 16 (𝑣 = (𝑧𝑢) → ((𝑥𝑛𝑛𝑣) ↔ (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
5453rexbidv 3182 . . . . . . . . . . . . . . 15 (𝑣 = (𝑧𝑢) → (∃𝑛𝑡 (𝑥𝑛𝑛𝑣) ↔ ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢))))
5551, 54imbi12d 333 . . . . . . . . . . . . . 14 (𝑣 = (𝑧𝑢) → ((𝑥𝑣 → ∃𝑛𝑡 (𝑥𝑛𝑛𝑣)) ↔ (𝑥 ∈ (𝑧𝑢) → ∃𝑛𝑡 (𝑥𝑛𝑛 ⊆ (𝑧𝑢)))))
5655rspcv 3437 . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . 18 ((((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) ∧ 𝑡 ∈ 𝒫 (𝑗t 𝑢)) ∧ 𝑛𝑡) → 𝑥𝑢)
59 elin 3931 . . . . . . . . . . . . . . . . . . 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 3970 . . . . . . . . . . . . . . . . . . 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 3976 . . . . . . . . . . . . . . . . . 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 3147 . . . . . . . . . . . . . . 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 3335 . . . . . . . . . . . . . . . . . 18 𝑛 ∈ V
7069inex1 4943 . . . . . . . . . . . . . . . . 17 (𝑛𝑢) ∈ V
7170rgenw 3054 . . . . . . . . . . . . . . . 16 𝑛𝑡 (𝑛𝑢) ∈ V
72 eleq2 2820 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑛𝑢) → (𝑥𝑤𝑥 ∈ (𝑛𝑢)))
73 sseq1 3759 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑛𝑢) → (𝑤𝑧 ↔ (𝑛𝑢) ⊆ 𝑧))
7472, 73anbi12d 749 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑛𝑢) → ((𝑥𝑤𝑤𝑧) ↔ (𝑥 ∈ (𝑛𝑢) ∧ (𝑛𝑢) ⊆ 𝑧)))
7529, 74rexrnmpt 6524 . . . . . . . . . . . . . . . 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 3096 . . . . . . . . 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 4799 . . . . . . . . . . 11 (𝑦 = ran (𝑎𝑡 ↦ (𝑎𝑢)) → (𝑦 ≼ ω ↔ ran (𝑎𝑡 ↦ (𝑎𝑢)) ≼ ω))
84 rexeq 3270 . . . . . . . . . . . . 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 3116 . . . . . . . . . . 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 3441 . . . . . . . . 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 1471 . . . . . . . 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 3165 . . . . . . 7 ((((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) ∧ 𝑢𝑗) ∧ (𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔)) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
91903adantr1 1172 . . . . . 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 20905 . . . . . . . 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 21471 . . . . . . 7 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑗𝑗𝑥 𝑗) → ∃𝑢𝑗 (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔))
9892, 95, 96, 97syl3anc 1473 . . . . . 6 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → ∃𝑢𝑗 (𝑢 𝑗𝑥𝑢 ∧ (𝑗t 𝑢) ∈ 1st𝜔))
9991, 98r19.29a 3208 . . . . 5 ((𝑗 ∈ Locally 1st𝜔 ∧ 𝑥 𝑗) → ∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
10099ralrimiva 3096 . . . 4 (𝑗 ∈ Locally 1st𝜔 → ∀𝑥 𝑗𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧𝑗 (𝑥𝑧 → ∃𝑤𝑦 (𝑥𝑤𝑤𝑧))))
1017is1stc2 21439 . . . 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 3740 . 2 Locally 1st𝜔 ⊆ 1st𝜔
104 1stcrest 21450 . . . . 5 ((𝑗 ∈ 1st𝜔 ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ 1st𝜔)
105104adantl 473 . . . 4 ((⊤ ∧ (𝑗 ∈ 1st𝜔 ∧ 𝑥𝑗)) → (𝑗t 𝑥) ∈ 1st𝜔)
106 1stctop 21440 . . . . . 6 (𝑗 ∈ 1st𝜔 → 𝑗 ∈ Top)
107106ssriv 3740 . . . . 5 1st𝜔 ⊆ Top
108107a1i 11 . . . 4 (⊤ → 1st𝜔 ⊆ Top)
109105, 108restlly 21480 . . 3 (⊤ → 1st𝜔 ⊆ Locally 1st𝜔)
110109trud 1634 . 2 1st𝜔 ⊆ Locally 1st𝜔
111103, 110eqssi 3752 1 Locally 1st𝜔 = 1st𝜔
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wtru 1625  wcel 2131  wral 3042  wrex 3043  Vcvv 3332  cin 3706  wss 3707  𝒫 cpw 4294   cuni 4580   class class class wbr 4796  cmpt 4873  ran crn 5259  wf 6037  (class class class)co 6805  ωcom 7222  cdom 8111  t crest 16275  Topctop 20892  1st𝜔c1stc 21434  Locally clly 21461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-oadd 7725  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-fin 8117  df-fi 8474  df-card 8947  df-acn 8950  df-rest 16277  df-topgen 16298  df-top 20893  df-topon 20910  df-bases 20944  df-1stc 21436  df-lly 21463
This theorem is referenced by:  dis1stc  21496
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