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Metamath Proof Explorer

Theorem eqvreldmqs

Description: Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 26-Sep-2021) (Revised by Peter Mazsa, 17-Jul-2023)

Ref Expression
Assertion eqvreldmqs ⊢ EqvRel ≀ E -1 ↾ A ∧ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A ↔ CoElEqvRel A ∧ ⋃ A / ∼ A = A

Proof

Step Hyp Ref Expression
1 df-coeleqvrel ⊢ CoElEqvRel A ↔ EqvRel ≀ E -1 ↾ A
2 1 bicomi ⊢ EqvRel ≀ E -1 ↾ A ↔ CoElEqvRel A
3 dmqs1cosscnvepreseq ⊢ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A ↔ ⋃ A / ∼ A = A
4 2 3 anbi12i ⊢ EqvRel ≀ E -1 ↾ A ∧ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A ↔ CoElEqvRel A ∧ ⋃ A / ∼ A = A