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Theorem ntrclsiex 37872
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (Contributed by RP, 21-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)
Assertion
Ref Expression
ntrclsiex (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝐼(𝑖,𝑗,𝑘)   𝐾(𝑖,𝑗,𝑘)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsiex
StepHypRef Expression
1 ntrcls.o . . . . 5 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . . . 5 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . . 5 (𝜑𝐼𝐷𝐾)
41, 2, 3ntrclsf1o 37870 . . . 4 (𝜑𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
5 f1orel 6107 . . . 4 (𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵) → Rel 𝐷)
64, 5syl 17 . . 3 (𝜑 → Rel 𝐷)
7 releldm 5328 . . 3 ((Rel 𝐷𝐼𝐷𝐾) → 𝐼 ∈ dom 𝐷)
86, 3, 7syl2anc 692 . 2 (𝜑𝐼 ∈ dom 𝐷)
9 f1odm 6108 . . 3 (𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵𝑚 𝒫 𝐵))
104, 9syl 17 . 2 (𝜑 → dom 𝐷 = (𝒫 𝐵𝑚 𝒫 𝐵))
118, 10eleqtrd 2700 1 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3190  cdif 3557  𝒫 cpw 4136   class class class wbr 4623  cmpt 4683  dom cdm 5084  Rel wrel 5089  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  𝑚 cmap 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-map 7819
This theorem is referenced by:  ntrclskex  37873  ntrclsfv1  37874  ntrclsfveq2  37880  ntrclscls00  37885  ntrclsiso  37886  ntrclsk2  37887  ntrclskb  37888  ntrclsk3  37889  ntrclsk13  37890  ntrclsk4  37891  clsneikex  37925
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