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Theorem ntrneineine1lem 37864
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 481 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5 ntrnei.x . . . . . 6 (𝜑𝑋𝐵)
65adantr 481 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
7 simpr 477 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
81, 2, 4, 6, 7ntrneiel 37861 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
98notbid 308 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (¬ 𝑋 ∈ (𝐼𝑠) ↔ ¬ 𝑠 ∈ (𝑁𝑋)))
109rexbidva 3042 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋)))
111, 2, 3ntrneinex 37857 . . . . . . 7 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
12 elmapi 7823 . . . . . . 7 (𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
1311, 12syl 17 . . . . . 6 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
1413, 5ffvelrnd 6316 . . . . 5 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1514elpwid 4141 . . . 4 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
16 biortn 421 . . . 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 2913 . . . 4 (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁𝑋)))
19 nss 3642 . . . 4 (¬ 𝒫 𝐵 ⊆ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁𝑋)))
2018, 19bitr4i 267 . . 3 (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋))
21 df-ne 2791 . . . 4 ((𝑁𝑋) ≠ 𝒫 𝐵 ↔ ¬ (𝑁𝑋) = 𝒫 𝐵)
22 ianor 509 . . . . 5 (¬ ((𝑁𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁𝑋)) ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
23 eqss 3598 . . . . 5 ((𝑁𝑋) = 𝒫 𝐵 ↔ ((𝑁𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁𝑋)))
2422, 23xchnxbir 323 . . . 4 (¬ (𝑁𝑋) = 𝒫 𝐵 ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
2521, 24bitri 264 . . 3 ((𝑁𝑋) ≠ 𝒫 𝐵 ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
2617, 20, 253bitr4g 303 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ (𝑁𝑋) ≠ 𝒫 𝐵))
2710, 26bitrd 268 1 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by:  ntrneineine1  37868
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