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Theorem ntrclsbex 43250
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then the base set exists. (Contributed by RP, 21-May-2021.)
Hypotheses
Ref Expression
ntrclsbex.d 𝐷 = (𝑂𝐵)
ntrclsbex.r (𝜑𝐼𝐷𝐾)
Assertion
Ref Expression
ntrclsbex (𝜑𝐵 ∈ V)

Proof of Theorem ntrclsbex
StepHypRef Expression
1 ntrclsbex.r . 2 (𝜑𝐼𝐷𝐾)
2 ntrclsbex.d . . 3 𝐷 = (𝑂𝐵)
32a1i 11 . 2 (𝜑𝐷 = (𝑂𝐵))
41, 3brfvimex 43242 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  Vcvv 3473   class class class wbr 5148  cfv 6543
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-nul 5306  ax-pr 5427
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-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551
This theorem is referenced by:  ntrclsrcomplex  43251  ntrclsf1o  43267  ntrclsnvobr  43268  ntrclselnel1  43273  ntrclsfv  43275  ntrclscls00  43282  ntrclsiso  43283  ntrclsk2  43284  ntrclskb  43285  ntrclsk3  43286  ntrclsk13  43287
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