Theorem ntrclsbex 44217

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 44209	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3438   class class class wbr 5096  ‘cfv 6490

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 2115  ax-9 2123  ax-ext 2706  ax-nul 5249  ax-pr 5375

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498

This theorem is referenced by:  ntrclsrcomplex  44218  ntrclsf1o  44234  ntrclsnvobr  44235  ntrclselnel1  44240  ntrclsfv  44242  ntrclscls00  44249  ntrclsiso  44250  ntrclsk2  44251  ntrclskb  44252  ntrclsk3  44253  ntrclsk13  44254