Theorem ntrclsbex 44571

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 44563	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453   class class class wbr 5097  ‘cfv 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524

This theorem is referenced by:  ntrclsrcomplex  44572  ntrclsf1o  44588  ntrclsnvobr  44589  ntrclselnel1  44594  ntrclsfv  44596  ntrclscls00  44603  ntrclsiso  44604  ntrclsk2  44605  ntrclskb  44606  ntrclsk3  44607  ntrclsk13  44608