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Theorem ntrclsbex 44024
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 44016 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478   class class class wbr 5148  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571
This theorem is referenced by:  ntrclsrcomplex  44025  ntrclsf1o  44041  ntrclsnvobr  44042  ntrclselnel1  44047  ntrclsfv  44049  ntrclscls00  44056  ntrclsiso  44057  ntrclsk2  44058  ntrclskb  44059  ntrclsk3  44060  ntrclsk13  44061
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