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Theorem ntrclsnvobr 42306
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then they are related the opposite way. (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
ntrclsnvobr (𝜑𝐾𝐷𝐼)
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 ntrclsnvobr
StepHypRef Expression
1 ntrcls.o . . 3 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . 3 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . 4 (𝜑𝐼𝐷𝐾)
42, 3ntrclsbex 42288 . . 3 (𝜑𝐵 ∈ V)
51, 2, 4dssmapnvod 42274 . 2 (𝜑𝐷 = 𝐷)
61, 2, 3ntrclsf1o 42305 . . . 4 (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
7 f1orel 6787 . . . 4 (𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵) → Rel 𝐷)
8 relbrcnvg 6057 . . . 4 (Rel 𝐷 → (𝐾𝐷𝐼𝐼𝐷𝐾))
96, 7, 83syl 18 . . 3 (𝜑 → (𝐾𝐷𝐼𝐼𝐷𝐾))
103, 9mpbird 256 . 2 (𝜑𝐾𝐷𝐼)
115, 10breqdi 5120 1 (𝜑𝐾𝐷𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  Vcvv 3445  cdif 3907  𝒫 cpw 4560   class class class wbr 5105  cmpt 5188  ccnv 5632  Rel wrel 5638  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7356  m cmap 8764
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7920  df-2nd 7921  df-map 8766
This theorem is referenced by:  ntrclskex  42308  ntrclsfv2  42310  ntrclselnel2  42312  ntrclsfveq2  42315  ntrclsk4  42326
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