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Theorem ntrclsiex 40743
 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 40741 . . . 4 (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
5 f1orel 6597 . . . 4 (𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵) → Rel 𝐷)
64, 5syl 17 . . 3 (𝜑 → Rel 𝐷)
7 releldm 5782 . . 3 ((Rel 𝐷𝐼𝐷𝐾) → 𝐼 ∈ dom 𝐷)
86, 3, 7syl2anc 587 . 2 (𝜑𝐼 ∈ dom 𝐷)
9 f1odm 6598 . . 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 2895 1 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  Vcvv 3444   ∖ cdif 3881  𝒫 cpw 4500   class class class wbr 5033   ↦ cmpt 5113  dom cdm 5523  Rel wrel 5528  –1-1-onto→wf1o 6327  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395 This theorem is referenced by:  ntrclskex  40744  ntrclsfv1  40745  ntrclsfveq2  40751  ntrclscls00  40756  ntrclsiso  40757  ntrclsk2  40758  ntrclskb  40759  ntrclsk3  40760  ntrclsk13  40761  ntrclsk4  40762  clsneikex  40796
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