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Theorem ntrclsbex 40724
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 40716 1 (𝜑𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  Vcvv 3444   class class class wbr 5033  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177  ax-pow 5234
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336
This theorem is referenced by:  ntrclsrcomplex  40725  ntrclsf1o  40741  ntrclsnvobr  40742  ntrclselnel1  40747  ntrclsfv  40749  ntrclscls00  40756  ntrclsiso  40757  ntrclsk2  40758  ntrclskb  40759  ntrclsk3  40760  ntrclsk13  40761
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