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Metamath Proof Explorer

Theorem eqvreldmqs2

Description: Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 30-Dec-2024)

Ref Expression
Assertion eqvreldmqs2 ⊢ EqvRel ≀ E -1 ↾ A ∧ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A ↔ EqvRel ∼ A ∧ ⋃ A / ∼ A = A

Proof

Step Hyp Ref Expression
1 df-coels ⊢ ∼ A = ≀ E -1 ↾ A
2 1 eqvreleqi ⊢ EqvRel ∼ A ↔ EqvRel ≀ E -1 ↾ A
3 2 bicomi ⊢ EqvRel ≀ E -1 ↾ A ↔ EqvRel ∼ A
4 dmqs1cosscnvepreseq ⊢ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A ↔ ⋃ A / ∼ A = A
5 3 4 anbi12i ⊢ EqvRel ≀ E -1 ↾ A ∧ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A ↔ EqvRel ∼ A ∧ ⋃ A / ∼ A = A