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Theorem ntrneineine1lem 38908
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 38905 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
98notbid 307 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (¬ 𝑋 ∈ (𝐼𝑠) ↔ ¬ 𝑠 ∈ (𝑁𝑋)))
109rexbidva 3197 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋)))
111, 2, 3ntrneinex 38901 . . . . . . 7 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
12 elmapi 8031 . . . . . . 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 4309 . . . 4 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
16 biortn 921 . . . 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 3067 . . . 4 (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁𝑋)))
19 nss 3812 . . . 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 2944 . . . 4 ((𝑁𝑋) ≠ 𝒫 𝐵 ↔ ¬ (𝑁𝑋) = 𝒫 𝐵)
22 ianor 962 . . . . 5 (¬ ((𝑁𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁𝑋)) ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
23 eqss 3767 . . . . 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 834   = wceq 1631  wex 1852  wcel 2145  wne 2943  wrex 3062  {crab 3065  Vcvv 3351  wss 3723  𝒫 cpw 4297   class class class wbr 4786  cmpt 4863  wf 6027  cfv 6031  (class class class)co 6793  cmpt2 6795  𝑚 cmap 8009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  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 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011
This theorem is referenced by:  ntrneineine1  38912
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