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Theorem ntrneiel 40782
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 6645 . . . . 5 (𝑚 = 𝑆 → (𝐼𝑚) = (𝐼𝑆))
32eleq2d 2875 . . . 4 (𝑚 = 𝑆 → (𝑋 ∈ (𝐼𝑚) ↔ 𝑋 ∈ (𝐼𝑆)))
43elrab3 3629 . . 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 40774 . . . . . 6 (𝜑𝐵 ∈ V)
109pwexd 5245 . . . . 5 (𝜑 → 𝒫 𝐵 ∈ V)
116, 7, 8ntrneiiex 40777 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
12 eqid 2798 . . . . 5 (𝐹𝐼) = (𝐹𝐼)
13 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
146, 10, 9, 7, 11, 12, 13fsovfvfvd 40710 . . . 4 (𝜑 → ((𝐹𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)})
156, 7, 8ntrneifv1 40780 . . . . 5 (𝜑 → (𝐹𝐼) = 𝑁)
1615fveq1d 6647 . . . 4 (𝜑 → ((𝐹𝐼)‘𝑋) = (𝑁𝑋))
1714, 16eqtr3d 2835 . . 3 (𝜑 → {𝑚 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑚)} = (𝑁𝑋))
1817eleq2d 2875 . 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 1538  wcel 2111  {crab 3110  Vcvv 3441  𝒫 cpw 4497   class class class wbr 5030  cmpt 5110  cfv 6324  (class class class)co 7135  cmpo 7137  m cmap 8389
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391
This theorem is referenced by:  ntrneifv3  40783  ntrneineine0lem  40784  ntrneineine1lem  40785  ntrneifv4  40786  ntrneiel2  40787  ntrneicls00  40790  ntrneicls11  40791  ntrneiiso  40792  ntrneik2  40793  ntrneix2  40794  ntrneikb  40795  ntrneixb  40796  ntrneik3  40797  ntrneix3  40798  ntrneik13  40799  ntrneix13  40800  ntrneik4w  40801  ntrneik4  40802  clsneiel1  40809  neicvgel1  40820
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