Theorem brredunds

Description: Binary relation on the class of all redundant sets. (Contributed by Peter Mazsa, 25-Oct-2022)

	Ref	Expression
Assertion	brredunds	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 Redunds ⟨ 𝐵 , 𝐶 ⟩ ↔ ( 𝐴 ⊆ 𝐵 ∧ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		sseq12	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵 ) )
2	1	3adant3	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵 ) )
3		ineq12	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑧 = 𝐶 ) → ( 𝑥 ∩ 𝑧 ) = ( 𝐴 ∩ 𝐶 ) )
4	3	3adant2	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝑥 ∩ 𝑧 ) = ( 𝐴 ∩ 𝐶 ) )
5		ineq12	⊢ ( ( 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝑦 ∩ 𝑧 ) = ( 𝐵 ∩ 𝐶 ) )
6	5	3adant1	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝑦 ∩ 𝑧 ) = ( 𝐵 ∩ 𝐶 ) )
7	4 6	eqeq12d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( ( 𝑥 ∩ 𝑧 ) = ( 𝑦 ∩ 𝑧 ) ↔ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) )
8	2 7	anbi12d	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( ( 𝑥 ⊆ 𝑦 ∧ ( 𝑥 ∩ 𝑧 ) = ( 𝑦 ∩ 𝑧 ) ) ↔ ( 𝐴 ⊆ 𝐵 ∧ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) ) )
9		df-redunds	⊢ Redunds = ◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ ( 𝑥 ⊆ 𝑦 ∧ ( 𝑥 ∩ 𝑧 ) = ( 𝑦 ∩ 𝑧 ) ) }
10	8 9	brcnvrabga	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 Redunds ⟨ 𝐵 , 𝐶 ⟩ ↔ ( 𝐴 ⊆ 𝐵 ∧ ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) ) ) )