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Theorem ntrclsfv1 38872
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (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)
Assertion
Ref Expression
ntrclsfv1 (𝜑 → (𝐷𝐼) = 𝐾)
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 ntrclsfv1
StepHypRef Expression
1 ntrcls.r . 2 (𝜑𝐼𝐷𝐾)
2 ntrcls.o . . . . . . 7 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
3 ntrcls.d . . . . . . 7 𝐷 = (𝑂𝐵)
42, 3, 1ntrclsf1o 38868 . . . . . 6 (𝜑𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
5 f1ofn 6279 . . . . . 6 (𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵) → 𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵))
64, 5syl 17 . . . . 5 (𝜑𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵))
72, 3, 1ntrclsiex 38870 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
86, 7jca 495 . . . 4 (𝜑 → (𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
9 fnfun 6128 . . . . . 6 (𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → Fun 𝐷)
109adantr 466 . . . . 5 ((𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → Fun 𝐷)
11 fndm 6130 . . . . . . 7 (𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵𝑚 𝒫 𝐵))
1211eleq2d 2835 . . . . . 6 (𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → (𝐼 ∈ dom 𝐷𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
1312biimpar 463 . . . . 5 ((𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → 𝐼 ∈ dom 𝐷)
1410, 13jca 495 . . . 4 ((𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → (Fun 𝐷𝐼 ∈ dom 𝐷))
158, 14syl 17 . . 3 (𝜑 → (Fun 𝐷𝐼 ∈ dom 𝐷))
16 funbrfvb 6379 . . 3 ((Fun 𝐷𝐼 ∈ dom 𝐷) → ((𝐷𝐼) = 𝐾𝐼𝐷𝐾))
1715, 16syl 17 . 2 (𝜑 → ((𝐷𝐼) = 𝐾𝐼𝐷𝐾))
181, 17mpbird 247 1 (𝜑 → (𝐷𝐼) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  cdif 3718  𝒫 cpw 4295   class class class wbr 4784  cmpt 4861  dom cdm 5249  Fun wfun 6025   Fn wfn 6026  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  𝑚 cmap 8008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010
This theorem is referenced by:  ntrclsfv2  38873  ntrclscls00  38883  ntrclsiso  38884  ntrclsk2  38885  ntrclskb  38886  ntrclsk3  38887  ntrclsk13  38888
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