Theorem ntrclsbex 38332

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 38324	1 ⊢ (𝜑 → 𝐵 ∈ V)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   class class class wbr 4653  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  ntrclsrcomplex  38333  ntrclsf1o  38349  ntrclsnvobr  38350  ntrclselnel1  38355  ntrclsfv  38357  ntrclscls00  38364  ntrclsiso  38365  ntrclsk2  38366  ntrclskb  38367  ntrclsk3  38368  ntrclsk13  38369