Theorem ntrclselnel1 44070

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 28-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)
ntrcls.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ntrcls.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
ntrclselnel1	⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐾,𝑘   𝑆,𝑗   𝜑,𝑖,𝑗,𝑘

Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝑆(𝑖,𝑘)   𝐼(𝑖,𝑗,𝑘)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrclselnel1

Step	Hyp	Ref	Expression
1		ntrcls.o	. . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . . . . . 7 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . . . . 7 ⊢ (𝜑 → 𝐼𝐷𝐾)
4	1, 2, 3	ntrclsfv2 44069	. . . . . 6 ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼)
5	4	eqcomd 2743	. . . . 5 ⊢ (𝜑 → 𝐼 = (𝐷‘𝐾))
6	5	fveq1d 6908	. . . 4 ⊢ (𝜑 → (𝐼‘𝑆) = ((𝐷‘𝐾)‘𝑆))
7	2, 3	ntrclsbex 44047	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
8	1, 2, 3	ntrclskex 44067	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
9		eqid 2737	. . . . 5 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾)
10		ntrcls.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
11		eqid 2737	. . . . 5 ⊢ ((𝐷‘𝐾)‘𝑆) = ((𝐷‘𝐾)‘𝑆)
12	1, 2, 7, 8, 9, 10, 11	dssmapfv3d 44032	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
13	6, 12	eqtrd 2777	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
14	13	eleq2d 2827	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))))
15		ntrcls.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
16		eldif 3961	. . . 4 ⊢ (𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))))
17	16	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆)))))
18	15, 17	mpbirand 707	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))))
19	14, 18	bitrd 279	1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∖ cdif 3948  𝒫 cpw 4600   class class class wbr 5143   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868

This theorem is referenced by:  ntrclselnel2  44071  clsneiel1  44121  neicvgel1  44132