Theorem ntrclsiex 43107

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (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
ntrclsiex	⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))

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 ntrclsiex

Step	Hyp	Ref	Expression
1		ntrcls.o	. . . . 5 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . . . 5 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . . 5 ⊢ (𝜑 → 𝐼𝐷𝐾)
4	1, 2, 3	ntrclsf1o 43105	. . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
5		f1orel 6836	. . . 4 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → Rel 𝐷)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → Rel 𝐷)
7		releldm 5943	. . 3 ⊢ ((Rel 𝐷 ∧ 𝐼𝐷𝐾) → 𝐼 ∈ dom 𝐷)
8	6, 3, 7	syl2anc 583	. 2 ⊢ (𝜑 → 𝐼 ∈ dom 𝐷)
9		f1odm 6837	. . 3 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵 ↑m 𝒫 𝐵))
10	4, 9	syl 17	. 2 ⊢ (𝜑 → dom 𝐷 = (𝒫 𝐵 ↑m 𝒫 𝐵))
11	8, 10	eleqtrd 2834	1 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ∖ cdif 3945  𝒫 cpw 4602   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  Rel wrel 5681  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7412   ↑_m cmap 8824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8826

This theorem is referenced by:  ntrclskex  43108  ntrclsfv1  43109  ntrclsfveq2  43115  ntrclscls00  43120  ntrclsiso  43121  ntrclsk2  43122  ntrclskb  43123  ntrclsk3  43124  ntrclsk13  43125  ntrclsk4  43126  clsneikex  43160