Theorem ntrclsiex 39121

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 39119	. . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵))
5		f1orel 6357	. . . 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 5560	. . 3 ⊢ ((Rel 𝐷 ∧ 𝐼𝐷𝐾) → 𝐼 ∈ dom 𝐷)
8	6, 3, 7	syl2anc 580	. 2 ⊢ (𝜑 → 𝐼 ∈ dom 𝐷)
9		f1odm 6358	. . 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 2878	1 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  Vcvv 3383   ∖ cdif 3764  𝒫 cpw 4347   class class class wbr 4841   ↦ cmpt 4920  dom cdm 5310  Rel wrel 5315  –1-1-onto→wf1o 6098  ‘cfv 6099  (class class class)co 6876   ↑_𝑚 cmap 8093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-map 8095

This theorem is referenced by:  ntrclskex  39122  ntrclsfv1  39123  ntrclsfveq2  39129  ntrclscls00  39134  ntrclsiso  39135  ntrclsk2  39136  ntrclskb  39137  ntrclsk3  39138  ntrclsk13  39139  ntrclsk4  39140  clsneikex  39174