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Theorem ntrclsiex 39121
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 39119 . . . 4 (𝜑𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
5 f1orel 6357 . . . 4 (𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵) → Rel 𝐷)
64, 5syl 17 . . 3 (𝜑 → Rel 𝐷)
7 releldm 5560 . . 3 ((Rel 𝐷𝐼𝐷𝐾) → 𝐼 ∈ dom 𝐷)
86, 3, 7syl2anc 580 . 2 (𝜑𝐼 ∈ dom 𝐷)
9 f1odm 6358 . . 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 2878 1 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  Vcvv 3383  cdif 3764  𝒫 cpw 4347   class class class wbr 4841  cmpt 4920  dom cdm 5310  Rel wrel 5315  1-1-ontowf1o 6098  cfv 6099  (class class class)co 6876  𝑚 cmap 8093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-map 8095
This theorem is referenced by:  ntrclskex  39122  ntrclsfv1  39123  ntrclsfveq2  39129  ntrclscls00  39134  ntrclsiso  39135  ntrclsk2  39136  ntrclskb  39137  ntrclsk3  39138  ntrclsk13  39139  ntrclsk4  39140  clsneikex  39174
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