Theorem ntrclsfv1 41665

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (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)

Assertion

Ref	Expression
ntrclsfv1	⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)

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 ntrclsfv1

Step	Hyp	Ref	Expression
1		ntrcls.r	. 2 ⊢ (𝜑 → 𝐼𝐷𝐾)
2		ntrcls.o	. . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
3		ntrcls.d	. . . . . . 7 ⊢ 𝐷 = (𝑂‘𝐵)
4	2, 3, 1	ntrclsf1o 41661	. . . . . 6 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
5		f1ofn 6717	. . . . . 6 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵))
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵))
7	2, 3, 1	ntrclsiex 41663	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
8	6, 7	jca 512	. . . 4 ⊢ (𝜑 → (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
9		fnfun 6533	. . . . . 6 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → Fun 𝐷)
10	9	adantr 481	. . . . 5 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → Fun 𝐷)
11		fndm 6536	. . . . . . 7 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵 ↑m 𝒫 𝐵))
12	11	eleq2d 2824	. . . . . 6 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → (𝐼 ∈ dom 𝐷 ↔ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)))
13	12	biimpar 478	. . . . 5 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → 𝐼 ∈ dom 𝐷)
14	10, 13	jca 512	. . . 4 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (Fun 𝐷 ∧ 𝐼 ∈ dom 𝐷))
15	8, 14	syl 17	. . 3 ⊢ (𝜑 → (Fun 𝐷 ∧ 𝐼 ∈ dom 𝐷))
16		funbrfvb 6824	. . 3 ⊢ ((Fun 𝐷 ∧ 𝐼 ∈ dom 𝐷) → ((𝐷‘𝐼) = 𝐾 ↔ 𝐼𝐷𝐾))
17	15, 16	syl 17	. 2 ⊢ (𝜑 → ((𝐷‘𝐼) = 𝐾 ↔ 𝐼𝐷𝐾))
18	1, 17	mpbird 256	1 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884  𝒫 cpw 4533   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589  Fun wfun 6427   Fn wfn 6428  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617

This theorem is referenced by:  ntrclsfv2  41666  ntrclscls00  41676  ntrclsiso  41677  ntrclsk2  41678  ntrclskb  41679  ntrclsk3  41680  ntrclsk13  41681