Theorem eccnvepres

Description: Restricted converse epsilon coset of B . (Contributed by Peter Mazsa, 11-Feb-2018) (Revised by Peter Mazsa, 21-Oct-2021)

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

Proof

Step	Hyp	Ref	Expression
1		brcnvep	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ◡ E 𝑥 ↔ 𝑥 ∈ 𝐵 ) )
2	1	anbi1cd	⊢ ( 𝐵 ∈ 𝑉 → ( ( 𝐵 ∈ 𝐴 ∧ 𝐵 ◡ E 𝑥 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) ) )
3	2	abbidv	⊢ ( 𝐵 ∈ 𝑉 → { 𝑥 ∣ ( 𝐵 ∈ 𝐴 ∧ 𝐵 ◡ E 𝑥 ) } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) } )
4		ecres	⊢ [ 𝐵 ] ( ◡ E ↾ 𝐴 ) = { 𝑥 ∣ ( 𝐵 ∈ 𝐴 ∧ 𝐵 ◡ E 𝑥 ) }
5		df-rab	⊢ { 𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) }
6	3 4 5	3eqtr4g	⊢ ( 𝐵 ∈ 𝑉 → [ 𝐵 ] ( ◡ E ↾ 𝐴 ) = { 𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴 } )