Theorem ntrclsnvobr 45244

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 45226	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
5	1, 2, 4	dssmapnvod 45212	. 2 ⊢ (𝜑 → ◡𝐷 = 𝐷)
6	1, 2, 3	ntrclsf1o 45243	. . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶1-1onto(𝒫 𝐵 ↑m 𝒫 𝐵))
7		f1orel 6849	. . . 4 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶1-1onto(𝒫 𝐵 ↑m 𝒫 𝐵) → Rel 𝐷)
8		relbrcnvg 6113	. . . 4 ⊢ (Rel 𝐷 → (𝐾◡𝐷𝐼 ↔ 𝐼𝐷𝐾))
9	6, 7, 8	3syl 19	. . 3 ⊢ (𝜑 → (𝐾◡𝐷𝐼 ↔ 𝐼𝐷𝐾))
10	3, 9	mpbird 260	. 2 ⊢ (𝜑 → 𝐾◡𝐷𝐼)
11	5, 10	breqdi 5121	1 ⊢ (𝜑 → 𝐾𝐷𝐼)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Vcvv 3454   ∖ cdif 4048  𝒫 cpw 4560   class class class wbr 5106   ↦ cmpt 5189  ^◡ccnv 5654  Rel wrel 5660  ⟶1-1ontowf1o 6557  ‘cfv 6558  (class class class)co 7443   ↑_m cmap 8875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-map 8877

This theorem is used by:  ntrclskex  45246  ntrclsfv2  45248  ntrclselnel2  45250  ntrclsfveq2  45253  ntrclsk4  45264