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Theorem ntrneiel 41309
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 6695 . . . . 5 (𝑚 = 𝑆 → (𝐼𝑚) = (𝐼𝑆))
32eleq2d 2816 . . . 4 (𝑚 = 𝑆 → (𝑋 ∈ (𝐼𝑚) ↔ 𝑋 ∈ (𝐼𝑆)))
43elrab3 3592 . . 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 41301 . . . . . 6 (𝜑𝐵 ∈ V)
109pwexd 5257 . . . . 5 (𝜑 → 𝒫 𝐵 ∈ V)
116, 7, 8ntrneiiex 41304 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
12 eqid 2736 . . . . 5 (𝐹𝐼) = (𝐹𝐼)
13 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
146, 10, 9, 7, 11, 12, 13fsovfvfvd 41237 . . . 4 (𝜑 → ((𝐹𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)})
156, 7, 8ntrneifv1 41307 . . . . 5 (𝜑 → (𝐹𝐼) = 𝑁)
1615fveq1d 6697 . . . 4 (𝜑 → ((𝐹𝐼)‘𝑋) = (𝑁𝑋))
1714, 16eqtr3d 2773 . . 3 (𝜑 → {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} = (𝑁𝑋))
1817eleq2d 2816 . 2 (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} ↔ 𝑆 ∈ (𝑁𝑋)))
195, 18bitr3d 284 1 (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2112  {crab 3055  Vcvv 3398  𝒫 cpw 4499   class class class wbr 5039  cmpt 5120  cfv 6358  (class class class)co 7191  cmpo 7193  m cmap 8486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-1st 7739  df-2nd 7740  df-map 8488
This theorem is referenced by:  ntrneifv3  41310  ntrneineine0lem  41311  ntrneineine1lem  41312  ntrneifv4  41313  ntrneiel2  41314  ntrneicls00  41317  ntrneicls11  41318  ntrneiiso  41319  ntrneik2  41320  ntrneix2  41321  ntrneikb  41322  ntrneixb  41323  ntrneik3  41324  ntrneix3  41325  ntrneik13  41326  ntrneix13  41327  ntrneik4w  41328  ntrneik4  41329  clsneiel1  41336  neicvgel1  41347
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