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Theorem ntrclsfv1 37832
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 37828 . . . . . 6 (𝜑𝐷:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵𝑚 𝒫 𝐵))
5 f1ofn 6095 . . . . . 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 37830 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
86, 7jca 554 . . . 4 (𝜑 → (𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
9 fnfun 5946 . . . . . 6 (𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → Fun 𝐷)
109adantr 481 . . . . 5 ((𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → Fun 𝐷)
11 fndm 5948 . . . . . . 7 (𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵𝑚 𝒫 𝐵))
1211eleq2d 2684 . . . . . 6 (𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) → (𝐼 ∈ dom 𝐷𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)))
1312biimpar 502 . . . . 5 ((𝐷 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵)) → 𝐼 ∈ dom 𝐷)
1410, 13jca 554 . . . 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 6195 . . 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 384   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673  dom cdm 5074  Fun wfun 5841   Fn wfn 5842  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  𝑚 cmap 7802
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by:  ntrclsfv2  37833  ntrclscls00  37843  ntrclsiso  37844  ntrclsk2  37845  ntrclskb  37846  ntrclsk3  37847  ntrclsk13  37848
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