Theorem ntrclselnel1 44048

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 44047	. . . . . 6 ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼)
5	4	eqcomd 2742	. . . . 5 ⊢ (𝜑 → 𝐼 = (𝐷‘𝐾))
6	5	fveq1d 6883	. . . 4 ⊢ (𝜑 → (𝐼‘𝑆) = ((𝐷‘𝐾)‘𝑆))
7	2, 3	ntrclsbex 44025	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
8	1, 2, 3	ntrclskex 44045	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
9		eqid 2736	. . . . 5 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾)
10		ntrcls.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
11		eqid 2736	. . . . 5 ⊢ ((𝐷‘𝐾)‘𝑆) = ((𝐷‘𝐾)‘𝑆)
12	1, 2, 7, 8, 9, 10, 11	dssmapfv3d 44010	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
13	6, 12	eqtrd 2771	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
14	13	eleq2d 2821	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))))
15		ntrcls.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
16		eldif 3941	. . . 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 2109  Vcvv 3464   ∖ cdif 3928  𝒫 cpw 4580   class class class wbr 5124   ↦ cmpt 5206  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8845

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-map 8847

This theorem is referenced by:  ntrclselnel2  44049  clsneiel1  44099  neicvgel1  44110