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Theorem ntrneixb 37413
 Description: The interiors (closures) of sets that span the base set also span the base set if and only if the neighborhoods (convergents) of every point contain at least one of every pair of sets that span the base set. (Contributed by RP, 11-Jun-2021.)
Hypotheses
Ref Expression
ntrnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
ntrnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r (𝜑𝐼𝐹𝑁)
Assertion
Ref Expression
ntrneixb (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠,𝑡,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠,𝑡,𝑥
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑥,𝑡,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑡,𝑖,𝑗,𝑠)   𝑁(𝑥,𝑡,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝑂(𝑥,𝑡,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)

Proof of Theorem ntrneixb
StepHypRef Expression
1 eqss 3583 . . . . . . . 8 (((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵 ↔ (((𝐼𝑠) ∪ (𝐼𝑡)) ⊆ 𝐵𝐵 ⊆ ((𝐼𝑠) ∪ (𝐼𝑡))))
21a1i 11 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵 ↔ (((𝐼𝑠) ∪ (𝐼𝑡)) ⊆ 𝐵𝐵 ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)))))
3 ntrnei.o . . . . . . . . . . . . . 14 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
4 ntrnei.f . . . . . . . . . . . . . 14 𝐹 = (𝒫 𝐵𝑂𝐵)
5 ntrnei.r . . . . . . . . . . . . . 14 (𝜑𝐼𝐹𝑁)
63, 4, 5ntrneiiex 37394 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
7 elmapi 7765 . . . . . . . . . . . . 13 (𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
86, 7syl 17 . . . . . . . . . . . 12 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
98ffvelrnda 6267 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
109elpwid 4118 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
1110adantr 480 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
128ffvelrnda 6267 . . . . . . . . . . 11 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐼𝑡) ∈ 𝒫 𝐵)
1312elpwid 4118 . . . . . . . . . 10 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐼𝑡) ⊆ 𝐵)
1413adantlr 747 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑡) ⊆ 𝐵)
1511, 14unssd 3751 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼𝑠) ∪ (𝐼𝑡)) ⊆ 𝐵)
1615biantrurd 528 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ (((𝐼𝑠) ∪ (𝐼𝑡)) ⊆ 𝐵𝐵 ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)))))
17 dfss3 3558 . . . . . . . . 9 (𝐵 ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)))
18 elun 3715 . . . . . . . . . 10 (𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)))
1918ralbii 2963 . . . . . . . . 9 (∀𝑥𝐵 𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)))
2017, 19bitri 263 . . . . . . . 8 (𝐵 ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)))
2120a1i 11 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡))))
222, 16, 213bitr2d 295 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵 ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡))))
2322imbi2d 329 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ((𝑠𝑡) = 𝐵 → ∀𝑥𝐵 (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)))))
24 r19.21v 2943 . . . . . 6 (∀𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡))) ↔ ((𝑠𝑡) = 𝐵 → ∀𝑥𝐵 (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡))))
2524a1i 11 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡))) ↔ ((𝑠𝑡) = 𝐵 → ∀𝑥𝐵 (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)))))
265ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
27 simpr 476 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
28 simpllr 795 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
293, 4, 26, 27, 28ntrneiel 37399 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
30 simplr 788 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
313, 4, 26, 27, 30ntrneiel 37399 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
3229, 31orbi12d 742 . . . . . . 7 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
3332imbi2d 329 . . . . . 6 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (((𝑠𝑡) = 𝐵 → (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡))) ↔ ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3433ralbidva 2968 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3523, 25, 343bitr2d 295 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3635ralbidva 2968 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3736ralbidva 2968 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
38 alrot3 2025 . . . 4 (∀𝑥𝑠𝑡((𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))) ↔ ∀𝑠𝑡𝑥((𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
39 3anrot 1036 . . . . . . 7 ((𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) ↔ (𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵))
4039imbi1i 338 . . . . . 6 (((𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))) ↔ ((𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
4140albii 1737 . . . . 5 (∀𝑥((𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))) ↔ ∀𝑥((𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
42412albii 1738 . . . 4 (∀𝑠𝑡𝑥((𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))) ↔ ∀𝑠𝑡𝑥((𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
4338, 42bitr2i 264 . . 3 (∀𝑠𝑡𝑥((𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))) ↔ ∀𝑥𝑠𝑡((𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
44 r3al 2924 . . 3 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑠𝑡𝑥((𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
45 r3al 2924 . . 3 (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝑠𝑡((𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵) → ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
4643, 44, 453bitr4i 291 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
4737, 46syl6bb 275 1 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by: (None)
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