Theorem br1cnvssrres

Description: Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019)

	Ref	Expression
Assertion	br1cnvssrres	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ◡ ( S ↾ 𝐴 ) 𝐶 ↔ ( 𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		relres	⊢ Rel ( S ↾ 𝐴 )
2	1	relbrcnv	⊢ ( 𝐵 ◡ ( S ↾ 𝐴 ) 𝐶 ↔ 𝐶 ( S ↾ 𝐴 ) 𝐵 )
3		brssrres	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐶 ( S ↾ 𝐴 ) 𝐵 ↔ ( 𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵 ) ) )
4	2 3	syl5bb	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ◡ ( S ↾ 𝐴 ) 𝐶 ↔ ( 𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵 ) ) )