Metamath Proof Explorer

Theorem dfcoeleqvrel

Description: Alternate definition of the coelement equivalence relation predicate: a coelement equivalence relation is an equivalence relation on coelements. Other alternate definitions should be based on eqvrelcoss2 , eqvrelcoss3 and eqvrelcoss4 when needed. (Contributed by Peter Mazsa, 28-Nov-2022)

Ref Expression
Assertion dfcoeleqvrel ⊢ CoElEqvRel A ↔ EqvRel ∼ A

Proof

Step Hyp Ref Expression
1 df-coeleqvrel ⊢ CoElEqvRel A ↔ EqvRel ≀ E -1 ↾ A
2 df-coels ⊢ ∼ A = ≀ E -1 ↾ A
3 2 eqvreleqi ⊢ EqvRel ∼ A ↔ EqvRel ≀ E -1 ↾ A
4 1 3 bitr4i ⊢ CoElEqvRel A ↔ EqvRel ∼ A