Description: Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023)
Ref | Expression | ||
---|---|---|---|
Assertion | elcoeleqvrels | ⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ CoElEqvRels ↔ ≀ ( ◡ E ↾ 𝐴 ) ∈ EqvRels ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseq2 | ⊢ ( 𝑎 = 𝐴 → ( ◡ E ↾ 𝑎 ) = ( ◡ E ↾ 𝐴 ) ) | |
2 | 1 | cosseqd | ⊢ ( 𝑎 = 𝐴 → ≀ ( ◡ E ↾ 𝑎 ) = ≀ ( ◡ E ↾ 𝐴 ) ) |
3 | 2 | eleq1d | ⊢ ( 𝑎 = 𝐴 → ( ≀ ( ◡ E ↾ 𝑎 ) ∈ EqvRels ↔ ≀ ( ◡ E ↾ 𝐴 ) ∈ EqvRels ) ) |
4 | df-coeleqvrels | ⊢ CoElEqvRels = { 𝑎 ∣ ≀ ( ◡ E ↾ 𝑎 ) ∈ EqvRels } | |
5 | 3 4 | elab2g | ⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ CoElEqvRels ↔ ≀ ( ◡ E ↾ 𝐴 ) ∈ EqvRels ) ) |