Theorem ntrclselnel1 44376

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 44375	. . . . . 6 ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼)
5	4	eqcomd 2743	. . . . 5 ⊢ (𝜑 → 𝐼 = (𝐷‘𝐾))
6	5	fveq1d 6837	. . . 4 ⊢ (𝜑 → (𝐼‘𝑆) = ((𝐷‘𝐾)‘𝑆))
7	2, 3	ntrclsbex 44353	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
8	1, 2, 3	ntrclskex 44373	. . . . 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 44338	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
13	6, 12	eqtrd 2772	. . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))))
14	13	eleq2d 2823	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑋 ∈ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))))
15		ntrcls.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
16		eldif 3912	. . . 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 708	. 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 1542   ∈ wcel 2114  Vcvv 3441   ∖ cdif 3899  𝒫 cpw 4555   class class class wbr 5099   ↦ cmpt 5180  ‘cfv 6493  (class class class)co 7361   ↑_m cmap 8768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-map 8770

This theorem is referenced by:  ntrclselnel2  44377  clsneiel1  44427  neicvgel1  44438