Theorem ec1cnvres

Description: Converse restricted coset of B . (Contributed by Peter Mazsa, 22-Mar-2019) (Revised by Peter Mazsa, 21-Oct-2021)

	Ref	Expression
Assertion	ec1cnvres	⊢ ( 𝐵 ∈ 𝑉 → [ 𝐵 ] ◡ ( 𝑅 ↾ 𝐴 ) = { 𝑥 ∈ 𝐴 ∣ 𝑥 𝑅 𝐵 } )

Proof

Step	Hyp	Ref	Expression
1		elec1cnvres	⊢ ( 𝐵 ∈ 𝑉 → ( 𝑥 ∈ [ 𝐵 ] ◡ ( 𝑅 ↾ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝐵 ) ) )
2	1	eqabdv	⊢ ( 𝐵 ∈ 𝑉 → [ 𝐵 ] ◡ ( 𝑅 ↾ 𝐴 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝐵 ) } )
3		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝑥 𝑅 𝐵 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 𝑅 𝐵 ) }
4	2 3	eqtr4di	⊢ ( 𝐵 ∈ 𝑉 → [ 𝐵 ] ◡ ( 𝑅 ↾ 𝐴 ) = { 𝑥 ∈ 𝐴 ∣ 𝑥 𝑅 𝐵 } )