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Theorem ntrneixb 40310
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 3986 . . . . . . . 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 40291 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
7 elmapi 8418 . . . . . . . . . . . . 13 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
86, 7syl 17 . . . . . . . . . . . 12 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
98ffvelrnda 6847 . . . . . . . . . . 11 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ∈ 𝒫 𝐵)
109elpwid 4556 . . . . . . . . . 10 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
1110adantr 481 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑠) ⊆ 𝐵)
128ffvelrnda 6847 . . . . . . . . . . 11 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐼𝑡) ∈ 𝒫 𝐵)
1312elpwid 4556 . . . . . . . . . 10 ((𝜑𝑡 ∈ 𝒫 𝐵) → (𝐼𝑡) ⊆ 𝐵)
1413adantlr 711 . . . . . . . . 9 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼𝑡) ⊆ 𝐵)
1511, 14unssd 4166 . . . . . . . 8 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼𝑠) ∪ (𝐼𝑡)) ⊆ 𝐵)
1615biantrurd 533 . . . . . . 7 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ (((𝐼𝑠) ∪ (𝐼𝑡)) ⊆ 𝐵𝐵 ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)))))
17 dfss3 3960 . . . . . . . . 9 (𝐵 ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)))
18 elun 4129 . . . . . . . . . 10 (𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)))
1918ralbii 3170 . . . . . . . . 9 (∀𝑥𝐵 𝑥 ∈ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)))
2017, 19bitri 276 . . . . . . . 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 308 . . . . . 6 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵 ↔ ∀𝑥𝐵 (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡))))
2322imbi2d 342 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ((𝑠𝑡) = 𝐵 → ∀𝑥𝐵 (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)))))
24 r19.21v 3180 . . . . . 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 726 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝐼𝐹𝑁)
27 simpr 485 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑥𝐵)
28 simpllr 772 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑠 ∈ 𝒫 𝐵)
293, 4, 26, 27, 28ntrneiel 40296 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑥)))
30 simplr 765 . . . . . . . . 9 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → 𝑡 ∈ 𝒫 𝐵)
313, 4, 26, 27, 30ntrneiel 40296 . . . . . . . 8 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ (𝐼𝑡) ↔ 𝑡 ∈ (𝑁𝑥)))
3229, 31orbi12d 914 . . . . . . 7 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → ((𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡)) ↔ (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
3332imbi2d 342 . . . . . 6 ((((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥𝐵) → (((𝑠𝑡) = 𝐵 → (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡))) ↔ ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3433ralbidva 3201 . . . . 5 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑥 ∈ (𝐼𝑠) ∨ 𝑥 ∈ (𝐼𝑡))) ↔ ∀𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3523, 25, 343bitr2d 308 . . . 4 (((𝜑𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3635ralbidva 3201 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
3736ralbidva 3201 . 2 (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))
38 ralrot3 3366 . 2 (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵𝑥𝐵 ((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥))))
3937, 38syl6bb 288 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 207  wa 396  wo 843   = wceq 1530  wcel 2107  wral 3143  {crab 3147  Vcvv 3500  cun 3938  wss 3940  𝒫 cpw 4542   class class class wbr 5063  cmpt 5143  wf 6348  cfv 6352  (class class class)co 7148  cmpo 7150  m cmap 8396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-map 8398
This theorem is referenced by: (None)
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