Theorem ntrclsfv

Description: The value of the interior (closure) expressed in terms of the closure (interior). (Contributed by RP, 25-Jun-2021)

	Ref	Expression
Hypotheses	ntrcls.o	⊢ 𝑂 = ( 𝑖 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑖 ↑m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖 ∖ 𝑗 ) ) ) ) ) )
	ntrcls.d	⊢ 𝐷 = ( 𝑂 ‘ 𝐵 )
	ntrcls.r	⊢ ( 𝜑 → 𝐼 𝐷 𝐾 )
	ntrclsfv.s	⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝐵 )
Assertion	ntrclsfv	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑆 ) = ( 𝐵 ∖ ( 𝐾 ‘ ( 𝐵 ∖ 𝑆 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ntrcls.o	⊢ 𝑂 = ( 𝑖 ∈ V ↦ ( 𝑘 ∈ ( 𝒫 𝑖 ↑m 𝒫 𝑖 ) ↦ ( 𝑗 ∈ 𝒫 𝑖 ↦ ( 𝑖 ∖ ( 𝑘 ‘ ( 𝑖 ∖ 𝑗 ) ) ) ) ) )
2		ntrcls.d	⊢ 𝐷 = ( 𝑂 ‘ 𝐵 )
3		ntrcls.r	⊢ ( 𝜑 → 𝐼 𝐷 𝐾 )
4		ntrclsfv.s	⊢ ( 𝜑 → 𝑆 ∈ 𝒫 𝐵 )
5	1 2 3	ntrclsfv2	⊢ ( 𝜑 → ( 𝐷 ‘ 𝐾 ) = 𝐼 )
6	5	fveq1d	⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝐾 ) ‘ 𝑆 ) = ( 𝐼 ‘ 𝑆 ) )
7	2 3	ntrclsbex	⊢ ( 𝜑 → 𝐵 ∈ V )
8	1 2 3	ntrclskex	⊢ ( 𝜑 → 𝐾 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )
9		eqid	⊢ ( 𝐷 ‘ 𝐾 ) = ( 𝐷 ‘ 𝐾 )
10		eqid	⊢ ( ( 𝐷 ‘ 𝐾 ) ‘ 𝑆 ) = ( ( 𝐷 ‘ 𝐾 ) ‘ 𝑆 )
11	1 2 7 8 9 4 10	dssmapfv3d	⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝐾 ) ‘ 𝑆 ) = ( 𝐵 ∖ ( 𝐾 ‘ ( 𝐵 ∖ 𝑆 ) ) ) )
12	6 11	eqtr3d	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑆 ) = ( 𝐵 ∖ ( 𝐾 ‘ ( 𝐵 ∖ 𝑆 ) ) ) )