Theorem ntrclsfveq2

Description: If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-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 )
	ntrclsfv.c	⊢ ( 𝜑 → 𝐶 ∈ 𝒫 𝐵 )
Assertion	ntrclsfveq2	⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) = 𝐶 ↔ ( 𝐾 ‘ 𝑆 ) = ( 𝐵 ∖ 𝐶 ) ) )

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		ntrclsfv.c	⊢ ( 𝜑 → 𝐶 ∈ 𝒫 𝐵 )
6	1 2 3	ntrclsiex	⊢ ( 𝜑 → 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) )
7		elmapi	⊢ ( 𝐼 ∈ ( 𝒫 𝐵 ↑m 𝒫 𝐵 ) → 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
8	6 7	syl	⊢ ( 𝜑 → 𝐼 : 𝒫 𝐵 ⟶ 𝒫 𝐵 )
9	2 3	ntrclsrcomplex	⊢ ( 𝜑 → ( 𝐵 ∖ 𝑆 ) ∈ 𝒫 𝐵 )
10	8 9	ffvelrnd	⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) ∈ 𝒫 𝐵 )
11	10	elpwid	⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) ⊆ 𝐵 )
12	5	elpwid	⊢ ( 𝜑 → 𝐶 ⊆ 𝐵 )
13		rcompleq	⊢ ( ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) = 𝐶 ↔ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) ) = ( 𝐵 ∖ 𝐶 ) ) )
14	11 12 13	syl2anc	⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) = 𝐶 ↔ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) ) = ( 𝐵 ∖ 𝐶 ) ) )
15	1 2 3	ntrclsnvobr	⊢ ( 𝜑 → 𝐾 𝐷 𝐼 )
16	1 2 15 4	ntrclsfv	⊢ ( 𝜑 → ( 𝐾 ‘ 𝑆 ) = ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) ) )
17	16	eqeq1d	⊢ ( 𝜑 → ( ( 𝐾 ‘ 𝑆 ) = ( 𝐵 ∖ 𝐶 ) ↔ ( 𝐵 ∖ ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) ) = ( 𝐵 ∖ 𝐶 ) ) )
18	14 17	bitr4d	⊢ ( 𝜑 → ( ( 𝐼 ‘ ( 𝐵 ∖ 𝑆 ) ) = 𝐶 ↔ ( 𝐾 ‘ 𝑆 ) = ( 𝐵 ∖ 𝐶 ) ) )