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Theorem ntrneineine1lem 44097
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-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)
ntrnei.x (𝜑𝑋𝐵)
Assertion
Ref Expression
ntrneineine1lem (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ 𝒫 𝐵))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠   𝑘,𝐼,𝑙,𝑚   𝑁,𝑠   𝑋,𝑙,𝑚,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑖,𝑗,𝑠)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrneineine1lem
StepHypRef Expression
1 ntrnei.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 ntrnei.f . . . . 5 𝐹 = (𝒫 𝐵𝑂𝐵)
3 ntrnei.r . . . . . 6 (𝜑𝐼𝐹𝑁)
43adantr 480 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5 ntrnei.x . . . . . 6 (𝜑𝑋𝐵)
65adantr 480 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
7 simpr 484 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
81, 2, 4, 6, 7ntrneiel 44094 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
98notbid 318 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (¬ 𝑋 ∈ (𝐼𝑠) ↔ ¬ 𝑠 ∈ (𝑁𝑋)))
109rexbidva 3177 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋)))
111, 2, 3ntrneinex 44090 . . . . . . 7 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
12 elmapi 8889 . . . . . . 7 (𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
1311, 12syl 17 . . . . . 6 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
1413, 5ffvelcdmd 7105 . . . . 5 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1514elpwid 4609 . . . 4 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
16 biortn 938 . . . 4 ((𝑁𝑋) ⊆ 𝒫 𝐵 → (¬ 𝒫 𝐵 ⊆ (𝑁𝑋) ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋))))
1715, 16syl 17 . . 3 (𝜑 → (¬ 𝒫 𝐵 ⊆ (𝑁𝑋) ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋))))
18 df-rex 3071 . . . 4 (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁𝑋)))
19 nss 4048 . . . 4 (¬ 𝒫 𝐵 ⊆ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁𝑋)))
2018, 19bitr4i 278 . . 3 (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋))
21 df-ne 2941 . . . 4 ((𝑁𝑋) ≠ 𝒫 𝐵 ↔ ¬ (𝑁𝑋) = 𝒫 𝐵)
22 ianor 984 . . . . 5 (¬ ((𝑁𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁𝑋)) ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
23 eqss 3999 . . . . 5 ((𝑁𝑋) = 𝒫 𝐵 ↔ ((𝑁𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁𝑋)))
2422, 23xchnxbir 333 . . . 4 (¬ (𝑁𝑋) = 𝒫 𝐵 ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
2521, 24bitri 275 . . 3 ((𝑁𝑋) ≠ 𝒫 𝐵 ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
2617, 20, 253bitr4g 314 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ (𝑁𝑋) ≠ 𝒫 𝐵))
2710, 26bitrd 279 1 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  wne 2940  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by:  ntrneineine1  44101
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