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Theorem ntrneiel 43783
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 6891 . . . . 5 (𝑚 = 𝑆 → (𝐼𝑚) = (𝐼𝑆))
32eleq2d 2812 . . . 4 (𝑚 = 𝑆 → (𝑋 ∈ (𝐼𝑚) ↔ 𝑋 ∈ (𝐼𝑆)))
43elrab3 3682 . . 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 43775 . . . . . 6 (𝜑𝐵 ∈ V)
109pwexd 5374 . . . . 5 (𝜑 → 𝒫 𝐵 ∈ V)
116, 7, 8ntrneiiex 43778 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
12 eqid 2726 . . . . 5 (𝐹𝐼) = (𝐹𝐼)
13 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
146, 10, 9, 7, 11, 12, 13fsovfvfvd 43713 . . . 4 (𝜑 → ((𝐹𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)})
156, 7, 8ntrneifv1 43781 . . . . 5 (𝜑 → (𝐹𝐼) = 𝑁)
1615fveq1d 6893 . . . 4 (𝜑 → ((𝐹𝐼)‘𝑋) = (𝑁𝑋))
1714, 16eqtr3d 2768 . . 3 (𝜑 → {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} = (𝑁𝑋))
1817eleq2d 2812 . 2 (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} ↔ 𝑆 ∈ (𝑁𝑋)))
195, 18bitr3d 280 1 (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  {crab 3420  Vcvv 3463  𝒫 cpw 4598   class class class wbr 5144  cmpt 5227  cfv 6544  (class class class)co 7414  cmpo 7416  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7993  df-2nd 7994  df-map 8847
This theorem is referenced by:  ntrneifv3  43784  ntrneineine0lem  43785  ntrneineine1lem  43786  ntrneifv4  43787  ntrneiel2  43788  ntrneicls00  43791  ntrneicls11  43792  ntrneiiso  43793  ntrneik2  43794  ntrneix2  43795  ntrneikb  43796  ntrneixb  43797  ntrneik3  43798  ntrneix3  43799  ntrneik13  43800  ntrneix13  43801  ntrneik4w  43802  ntrneik4  43803  clsneiel1  43810  neicvgel1  43821
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