Theorem ntrclsnvobr 44045

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 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)

Assertion

Ref	Expression
ntrclsnvobr	⊢ (𝜑 → 𝐾𝐷𝐼)

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

Step	Hyp	Ref	Expression
1		ntrcls.o	. . 3 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . 3 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . 4 ⊢ (𝜑 → 𝐼𝐷𝐾)
4	2, 3	ntrclsbex 44027	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
5	1, 2, 4	dssmapnvod 44013	. 2 ⊢ (𝜑 → ◡𝐷 = 𝐷)
6	1, 2, 3	ntrclsf1o 44044	. . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
7		f1orel 6767	. . . 4 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → Rel 𝐷)
8		relbrcnvg 6056	. . . 4 ⊢ (Rel 𝐷 → (𝐾◡𝐷𝐼 ↔ 𝐼𝐷𝐾))
9	6, 7, 8	3syl 18	. . 3 ⊢ (𝜑 → (𝐾◡𝐷𝐼 ↔ 𝐼𝐷𝐾))
10	3, 9	mpbird 257	. 2 ⊢ (𝜑 → 𝐾◡𝐷𝐼)
11	5, 10	breqdi 5107	1 ⊢ (𝜑 → 𝐾𝐷𝐼)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  Vcvv 3436   ∖ cdif 3900  𝒫 cpw 4551   class class class wbr 5092   ↦ cmpt 5173  ^◡ccnv 5618  Rel wrel 5624  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-map 8755

This theorem is referenced by:  ntrclskex  44047  ntrclsfv2  44049  ntrclselnel2  44051  ntrclsfveq2  44054  ntrclsk4  44065