Theorem ntrclsbex 44047

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 44039	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3480   class class class wbr 5143  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569

This theorem is referenced by:  ntrclsrcomplex  44048  ntrclsf1o  44064  ntrclsnvobr  44065  ntrclselnel1  44070  ntrclsfv  44072  ntrclscls00  44079  ntrclsiso  44080  ntrclsk2  44081  ntrclskb  44082  ntrclsk3  44083  ntrclsk13  44084