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Theorem ntrneiel 37399
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 6103 . . . . 5 (𝑚 = 𝑆 → (𝐼𝑚) = (𝐼𝑆))
32eleq2d 2673 . . . 4 (𝑚 = 𝑆 → (𝑋 ∈ (𝐼𝑚) ↔ 𝑋 ∈ (𝐼𝑆)))
43elrab3 3332 . . 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 37391 . . . . . 6 (𝜑𝐵 ∈ V)
10 pwexg 4776 . . . . . 6 (𝐵 ∈ V → 𝒫 𝐵 ∈ V)
119, 10syl 17 . . . . 5 (𝜑 → 𝒫 𝐵 ∈ V)
126, 7, 8ntrneiiex 37394 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
13 eqid 2610 . . . . 5 (𝐹𝐼) = (𝐹𝐼)
14 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
156, 11, 9, 7, 12, 13, 14fsovfvfvd 37325 . . . 4 (𝜑 → ((𝐹𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)})
166, 7, 8ntrneifv1 37397 . . . . 5 (𝜑 → (𝐹𝐼) = 𝑁)
1716fveq1d 6105 . . . 4 (𝜑 → ((𝐹𝐼)‘𝑋) = (𝑁𝑋))
1815, 17eqtr3d 2646 . . 3 (𝜑 → {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} = (𝑁𝑋))
1918eleq2d 2673 . 2 (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} ↔ 𝑆 ∈ (𝑁𝑋)))
205, 19bitr3d 269 1 (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wcel 1977  {crab 2900  Vcvv 3173  𝒫 cpw 4108   class class class wbr 4583  cmpt 4643  cfv 5804  (class class class)co 6549  cmpt2 6551  𝑚 cmap 7744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746
This theorem is referenced by:  ntrneifv3  37400  ntrneineine0lem  37401  ntrneineine1lem  37402  ntrneifv4  37403  ntrneiel2  37404  ntrneicls00  37407  ntrneicls11  37408  ntrneiiso  37409  ntrneik2  37410  ntrneix2  37411  ntrneikb  37412  ntrneixb  37413  ntrneik3  37414  ntrneix3  37415  ntrneik13  37416  ntrneix13  37417  ntrneik4w  37418  ntrneik4  37419  clsneiel1  37426  neicvgel1  37437
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