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Theorem ntrclsbex 43361
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 43353 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468   class class class wbr 5141  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545
This theorem is referenced by:  ntrclsrcomplex  43362  ntrclsf1o  43378  ntrclsnvobr  43379  ntrclselnel1  43384  ntrclsfv  43386  ntrclscls00  43393  ntrclsiso  43394  ntrclsk2  43395  ntrclskb  43396  ntrclsk3  43397  ntrclsk13  43398
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