Theorem ntrclselnel1 44638

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 44637	. . . . . 6 ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼)
5	4	eqcomd 2769	. . . . 5 ⊢ (𝜑 → 𝐼 = (𝐷‘𝐾))
6	5	fveq1d 6869	. . . 4 ⊢ (𝜑 → (𝐼‘𝑆) = ((𝐷‘𝐾)‘𝑆))
7	2, 3	ntrclsbex 44615	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
8	1, 2, 3	ntrclskex 44635	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
9		eqid 2763	. . . . 5 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾)
10		ntrcls.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
11		eqid 2763	. . . . 5 ⊢ ((𝐷‘𝐾)‘𝑆) = ((𝐷‘𝐾)‘𝑆)
12	1, 2, 7, 8, 9, 10, 11	dssmapfv3d 44600	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
13	6, 12	eqtrd 2798	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
14	13	eleq2d 2849	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))))
15		ntrcls.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
16		eldif 3915	. . . 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 717	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))))
19	14, 18	bitrd 281	1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑆))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902  𝒫 cpw 4556   class class class wbr 5101   ↦ cmpt 5182  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810

This theorem is referenced by:  ntrclselnel2  44639  clsneiel1  44689  neicvgel1  44700