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Theorem ntrneineine1lem 40774
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 484 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
5 ntrnei.x . . . . . 6 (𝜑𝑋𝐵)
65adantr 484 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
7 simpr 488 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
81, 2, 4, 6, 7ntrneiel 40771 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
98notbid 321 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (¬ 𝑋 ∈ (𝐼𝑠) ↔ ¬ 𝑠 ∈ (𝑁𝑋)))
109rexbidva 3258 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋)))
111, 2, 3ntrneinex 40767 . . . . . . 7 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
12 elmapi 8415 . . . . . . 7 (𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
1311, 12syl 17 . . . . . 6 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
1413, 5ffvelrnd 6833 . . . . 5 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
1514elpwid 4511 . . . 4 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
16 biortn 935 . . . 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 3115 . . . 4 (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁𝑋)))
19 nss 3980 . . . 4 (¬ 𝒫 𝐵 ⊆ (𝑁𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ ¬ 𝑠 ∈ (𝑁𝑋)))
2018, 19bitr4i 281 . . 3 (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋))
21 df-ne 2991 . . . 4 ((𝑁𝑋) ≠ 𝒫 𝐵 ↔ ¬ (𝑁𝑋) = 𝒫 𝐵)
22 ianor 979 . . . . 5 (¬ ((𝑁𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁𝑋)) ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
23 eqss 3933 . . . . 5 ((𝑁𝑋) = 𝒫 𝐵 ↔ ((𝑁𝑋) ⊆ 𝒫 𝐵 ∧ 𝒫 𝐵 ⊆ (𝑁𝑋)))
2422, 23xchnxbir 336 . . . 4 (¬ (𝑁𝑋) = 𝒫 𝐵 ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
2521, 24bitri 278 . . 3 ((𝑁𝑋) ≠ 𝒫 𝐵 ↔ (¬ (𝑁𝑋) ⊆ 𝒫 𝐵 ∨ ¬ 𝒫 𝐵 ⊆ (𝑁𝑋)))
2617, 20, 253bitr4g 317 . 2 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑠 ∈ (𝑁𝑋) ↔ (𝑁𝑋) ≠ 𝒫 𝐵))
2710, 26bitrd 282 1 (𝜑 → (∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐼𝑠) ↔ (𝑁𝑋) ≠ 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wex 1781  wcel 2112  wne 2990  wrex 3110  {crab 3113  Vcvv 3444  wss 3884  𝒫 cpw 4500   class class class wbr 5033  cmpt 5113  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  m cmap 8393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395
This theorem is referenced by:  ntrneineine1  40778
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