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 } ) |
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 } ) |