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Theorem ntrclsfv1 43109
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 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
ntrcls.d 𝐷 = (𝑂𝐵)
ntrcls.r (𝜑𝐼𝐷𝐾)
Assertion
Ref Expression
ntrclsfv1 (𝜑 → (𝐷𝐼) = 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
StepHypRef Expression
1 ntrcls.r . 2 (𝜑𝐼𝐷𝐾)
2 ntrcls.o . . . . . . 7 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
3 ntrcls.d . . . . . . 7 𝐷 = (𝑂𝐵)
42, 3, 1ntrclsf1o 43105 . . . . . 6 (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
5 f1ofn 6834 . . . . . 6 (𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵) → 𝐷 Fn (𝒫 𝐵m 𝒫 𝐵))
64, 5syl 17 . . . . 5 (𝜑𝐷 Fn (𝒫 𝐵m 𝒫 𝐵))
72, 3, 1ntrclsiex 43107 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
86, 7jca 511 . . . 4 (𝜑 → (𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)))
9 fnfun 6649 . . . . . 6 (𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) → Fun 𝐷)
109adantr 480 . . . . 5 ((𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → Fun 𝐷)
11 fndm 6652 . . . . . . 7 (𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵m 𝒫 𝐵))
1211eleq2d 2818 . . . . . 6 (𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) → (𝐼 ∈ dom 𝐷𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)))
1312biimpar 477 . . . . 5 ((𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → 𝐼 ∈ dom 𝐷)
1410, 13jca 511 . . . 4 ((𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵)) → (Fun 𝐷𝐼 ∈ dom 𝐷))
158, 14syl 17 . . 3 (𝜑 → (Fun 𝐷𝐼 ∈ dom 𝐷))
16 funbrfvb 6946 . . 3 ((Fun 𝐷𝐼 ∈ dom 𝐷) → ((𝐷𝐼) = 𝐾𝐼𝐷𝐾))
1715, 16syl 17 . 2 (𝜑 → ((𝐷𝐼) = 𝐾𝐼𝐷𝐾))
181, 17mpbird 257 1 (𝜑 → (𝐷𝐼) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  cdif 3945  𝒫 cpw 4602   class class class wbr 5148  cmpt 5231  dom cdm 5676  Fun wfun 6537   Fn wfn 6538  1-1-ontowf1o 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:  ntrclsfv2  43110  ntrclscls00  43120  ntrclsiso  43121  ntrclsk2  43122  ntrclskb  43123  ntrclsk3  43124  ntrclsk13  43125
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