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Theorem ntrneiel 43408
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 29-May-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)
ntrnei.s (𝜑𝑆 ∈ 𝒫 𝐵)
Assertion
Ref Expression
ntrneiel (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑋)))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝑘,𝐼,𝑙,𝑚   𝑆,𝑚   𝑋,𝑙,𝑚   𝜑,𝑖,𝑗,𝑘,𝑙
Allowed substitution hints:   𝜑(𝑚)   𝑆(𝑖,𝑗,𝑘,𝑙)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrneiel
StepHypRef Expression
1 ntrnei.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
2 fveq2 6885 . . . . 5 (𝑚 = 𝑆 → (𝐼𝑚) = (𝐼𝑆))
32eleq2d 2813 . . . 4 (𝑚 = 𝑆 → (𝑋 ∈ (𝐼𝑚) ↔ 𝑋 ∈ (𝐼𝑆)))
43elrab3 3679 . . 3 (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} ↔ 𝑋 ∈ (𝐼𝑆)))
51, 4syl 17 . 2 (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} ↔ 𝑋 ∈ (𝐼𝑆)))
6 ntrnei.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
7 ntrnei.f . . . . . . 7 𝐹 = (𝒫 𝐵𝑂𝐵)
8 ntrnei.r . . . . . . 7 (𝜑𝐼𝐹𝑁)
96, 7, 8ntrneibex 43400 . . . . . 6 (𝜑𝐵 ∈ V)
109pwexd 5370 . . . . 5 (𝜑 → 𝒫 𝐵 ∈ V)
116, 7, 8ntrneiiex 43403 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
12 eqid 2726 . . . . 5 (𝐹𝐼) = (𝐹𝐼)
13 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
146, 10, 9, 7, 11, 12, 13fsovfvfvd 43338 . . . 4 (𝜑 → ((𝐹𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)})
156, 7, 8ntrneifv1 43406 . . . . 5 (𝜑 → (𝐹𝐼) = 𝑁)
1615fveq1d 6887 . . . 4 (𝜑 → ((𝐹𝐼)‘𝑋) = (𝑁𝑋))
1714, 16eqtr3d 2768 . . 3 (𝜑 → {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} = (𝑁𝑋))
1817eleq2d 2813 . 2 (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} ↔ 𝑆 ∈ (𝑁𝑋)))
195, 18bitr3d 281 1 (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  {crab 3426  Vcvv 3468  𝒫 cpw 4597   class class class wbr 5141  cmpt 5224  cfv 6537  (class class class)co 7405  cmpo 7407  m cmap 8822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824
This theorem is referenced by:  ntrneifv3  43409  ntrneineine0lem  43410  ntrneineine1lem  43411  ntrneifv4  43412  ntrneiel2  43413  ntrneicls00  43416  ntrneicls11  43417  ntrneiiso  43418  ntrneik2  43419  ntrneix2  43420  ntrneikb  43421  ntrneixb  43422  ntrneik3  43423  ntrneix3  43424  ntrneik13  43425  ntrneix13  43426  ntrneik4w  43427  ntrneik4  43428  clsneiel1  43435  neicvgel1  43446
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