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Theorem ntrclsbex 39102
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 39094 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  Vcvv 3383   class class class wbr 4841  cfv 6099
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-nul 4981  ax-pow 5033
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-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-iota 6062  df-fv 6107
This theorem is referenced by:  ntrclsrcomplex  39103  ntrclsf1o  39119  ntrclsnvobr  39120  ntrclselnel1  39125  ntrclsfv  39127  ntrclscls00  39134  ntrclsiso  39135  ntrclsk2  39136  ntrclskb  39137  ntrclsk3  39138  ntrclsk13  39139
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