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Theorem ntrclsfv1 43550
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 43546 . . . . . 6 (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
5 f1ofn 6835 . . . . . 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 43548 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
86, 7jca 510 . . . 4 (𝜑 → (𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)))
9 fnfun 6649 . . . . . 6 (𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) → Fun 𝐷)
109adantr 479 . . . . 5 ((𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → Fun 𝐷)
11 fndm 6652 . . . . . . 7 (𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵m 𝒫 𝐵))
1211eleq2d 2811 . . . . . 6 (𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) → (𝐼 ∈ dom 𝐷𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)))
1312biimpar 476 . . . . 5 ((𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → 𝐼 ∈ dom 𝐷)
1410, 13jca 510 . . . 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 6947 . . 3 ((Fun 𝐷𝐼 ∈ dom 𝐷) → ((𝐷𝐼) = 𝐾𝐼𝐷𝐾))
1715, 16syl 17 . 2 (𝜑 → ((𝐷𝐼) = 𝐾𝐼𝐷𝐾))
181, 17mpbird 256 1 (𝜑 → (𝐷𝐼) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  Vcvv 3463  cdif 3936  𝒫 cpw 4598   class class class wbr 5143  cmpt 5226  dom cdm 5672  Fun wfun 6537   Fn wfn 6538  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7416  m cmap 8843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-map 8845
This theorem is referenced by:  ntrclsfv2  43551  ntrclscls00  43561  ntrclsiso  43562  ntrclsk2  43563  ntrclskb  43564  ntrclsk3  43565  ntrclsk13  43566
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