This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dfcoels

Description: Alternate definition of the class of coelements on the class A . (Contributed by Peter Mazsa, 20-Apr-2019)

Ref Expression
Assertion dfcoels ∼ 𝐴 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢 ) }

Proof

Step Hyp Ref Expression
1 df-coels ⊢ ∼ 𝐴 = ≀ ( ◡ E ↾ 𝐴 )
2 1cossres ⊢ ≀ ( ◡ E ↾ 𝐴 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑢 ◡ E 𝑥 ∧ 𝑢 ◡ E 𝑦 ) }
3 brcnvep ⊢ ( 𝑢 ∈ V → ( 𝑢 ◡ E 𝑥 ↔ 𝑥 ∈ 𝑢 ) )
4 3 elv ⊢ ( 𝑢 ◡ E 𝑥 ↔ 𝑥 ∈ 𝑢 )
5 brcnvep ⊢ ( 𝑢 ∈ V → ( 𝑢 ◡ E 𝑦 ↔ 𝑦 ∈ 𝑢 ) )
6 5 elv ⊢ ( 𝑢 ◡ E 𝑦 ↔ 𝑦 ∈ 𝑢 )
7 4 6 anbi12i ⊢ ( ( 𝑢 ◡ E 𝑥 ∧ 𝑢 ◡ E 𝑦 ) ↔ ( 𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢 ) )
8 7 rexbii ⊢ ( ∃ 𝑢 ∈ 𝐴 ( 𝑢 ◡ E 𝑥 ∧ 𝑢 ◡ E 𝑦 ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢 ) )
9 8 opabbii ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑢 ◡ E 𝑥 ∧ 𝑢 ◡ E 𝑦 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢 ) }
10 1 2 9 3eqtri ⊢ ∼ 𝐴 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢 ) }