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Theorem ntrclselnel1 44070
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator 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, 28-May-2021.)
Hypotheses
Ref Expression
ntrcls.o 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
ntrcls.d 𝐷 = (𝑂𝐵)
ntrcls.r (𝜑𝐼𝐷𝐾)
ntrcls.x (𝜑𝑋𝐵)
ntrcls.s (𝜑𝑆 ∈ 𝒫 𝐵)
Assertion
Ref Expression
ntrclselnel1 (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑆))))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐾,𝑘   𝑆,𝑗   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝑆(𝑖,𝑘)   𝐼(𝑖,𝑗,𝑘)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrclselnel1
StepHypRef Expression
1 ntrcls.o . . . . . . 7 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . . . . . 7 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . . . . 7 (𝜑𝐼𝐷𝐾)
41, 2, 3ntrclsfv2 44069 . . . . . 6 (𝜑 → (𝐷𝐾) = 𝐼)
54eqcomd 2743 . . . . 5 (𝜑𝐼 = (𝐷𝐾))
65fveq1d 6908 . . . 4 (𝜑 → (𝐼𝑆) = ((𝐷𝐾)‘𝑆))
72, 3ntrclsbex 44047 . . . . 5 (𝜑𝐵 ∈ V)
81, 2, 3ntrclskex 44067 . . . . 5 (𝜑𝐾 ∈ (𝒫 𝐵m 𝒫 𝐵))
9 eqid 2737 . . . . 5 (𝐷𝐾) = (𝐷𝐾)
10 ntrcls.s . . . . 5 (𝜑𝑆 ∈ 𝒫 𝐵)
11 eqid 2737 . . . . 5 ((𝐷𝐾)‘𝑆) = ((𝐷𝐾)‘𝑆)
121, 2, 7, 8, 9, 10, 11dssmapfv3d 44032 . . . 4 (𝜑 → ((𝐷𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
136, 12eqtrd 2777 . . 3 (𝜑 → (𝐼𝑆) = (𝐵 ∖ (𝐾‘(𝐵𝑆))))
1413eleq2d 2827 . 2 (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ 𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵𝑆)))))
15 ntrcls.x . . 3 (𝜑𝑋𝐵)
16 eldif 3961 . . . 4 (𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵𝑆))) ↔ (𝑋𝐵 ∧ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑆))))
1716a1i 11 . . 3 (𝜑 → (𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵𝑆))) ↔ (𝑋𝐵 ∧ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑆)))))
1815, 17mpbirand 707 . 2 (𝜑 → (𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵𝑆))) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑆))))
1914, 18bitrd 279 1 (𝜑 → (𝑋 ∈ (𝐼𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by:  ntrclselnel2  44071  clsneiel1  44121  neicvgel1  44132
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