Theorem elecres

Description: Elementhood in the restricted coset of B . (Contributed by Peter Mazsa, 21-Sep-2018)

	Ref	Expression
Assertion	elecres	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐶 ∈ [ 𝐵 ] ( 𝑅 ↾ 𝐴 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		relres	⊢ Rel ( 𝑅 ↾ 𝐴 )
2		relelec	⊢ ( Rel ( 𝑅 ↾ 𝐴 ) → ( 𝐶 ∈ [ 𝐵 ] ( 𝑅 ↾ 𝐴 ) ↔ 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 ) )
3	1 2	ax-mp	⊢ ( 𝐶 ∈ [ 𝐵 ] ( 𝑅 ↾ 𝐴 ) ↔ 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 )
4		brres	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐵 ( 𝑅 ↾ 𝐴 ) 𝐶 ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) ) )
5	3 4	bitrid	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐶 ∈ [ 𝐵 ] ( 𝑅 ↾ 𝐴 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 𝑅 𝐶 ) ) )