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Theorem ntrneineine1lem 38901
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 466 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5 ntrnei.x . . . . . 6 (𝜑𝑋𝐵)
65adantr 466 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
7 simpr 471 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
81, 2, 4, 6, 7ntrneiel 38898 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
98notbid 307 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (¬ 𝑋 ∈ (𝐼𝑠) ↔ ¬ 𝑠 ∈ (𝑁𝑋)))
109rexbidva 3196 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋)))
111, 2, 3ntrneinex 38894 . . . . . . 7 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
12 elmapi 8030 . . . . . . 7 (𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
1311, 12syl 17 . . . . . 6 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
1413, 5ffvelrnd 6503 . . . . 5 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1514elpwid 4307 . . . 4 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
16 biortn 897 . . . 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 3066 . . . 4 (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁𝑋)))
19 nss 3810 . . . 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 2943 . . . 4 ((𝑁𝑋) ≠ 𝒫 𝐵 ↔ ¬ (𝑁𝑋) = 𝒫 𝐵)
22 ianor 910 . . . . 5 (¬ ((𝑁𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁𝑋)) ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
23 eqss 3765 . . . . 5 ((𝑁𝑋) = 𝒫 𝐵 ↔ ((𝑁𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁𝑋)))
2422, 23xchnxbir 322 . . . 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  wa 382  wo 826   = wceq 1630  wex 1851  wcel 2144  wne 2942  wrex 3061  {crab 3064  Vcvv 3349  wss 3721  𝒫 cpw 4295   class class class wbr 4784  cmpt 4861  wf 6027  cfv 6031  (class class class)co 6792  cmpt2 6794  𝑚 cmap 8008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010
This theorem is referenced by:  ntrneineine1  38905
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