Theorem ntrclsbex 44067

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 = (𝑂‘𝐵)
ntrclsbex.r	⊢ (𝜑 → 𝐼𝐷𝐾)

Assertion

Ref	Expression
ntrclsbex	⊢ (𝜑 → 𝐵 ∈ V)

Proof of Theorem ntrclsbex

Step	Hyp	Ref	Expression
1		ntrclsbex.r	. 2 ⊢ (𝜑 → 𝐼𝐷𝐾)
2		ntrclsbex.d	. . 3 ⊢ 𝐷 = (𝑂‘𝐵)
3	2	a1i 11	. 2 ⊢ (𝜑 → 𝐷 = (𝑂‘𝐵))
4	1, 3	brfvimex 44059	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436   class class class wbr 5086  ‘cfv 6476

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484

This theorem is referenced by:  ntrclsrcomplex  44068  ntrclsf1o  44084  ntrclsnvobr  44085  ntrclselnel1  44090  ntrclsfv  44092  ntrclscls00  44099  ntrclsiso  44100  ntrclsk2  44101  ntrclskb  44102  ntrclsk3  44103  ntrclsk13  44104