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Metamath Proof Explorer

Theorem dmqscoelseq

Description: Two ways to express the equality of the domain quotient of the coelements on the class A with the class A . (Contributed by Peter Mazsa, 26-Sep-2021)

Ref Expression
Assertion dmqscoelseq ⊢ dom ⁡ ∼ A / ∼ A = A ↔ ⋃ A / ∼ A = A

Proof

Step Hyp Ref Expression
1 dmcoels ⊢ dom ⁡ ∼ A = ⋃ A
2 1 qseq1i ⊢ dom ⁡ ∼ A / ∼ A = ⋃ A / ∼ A
3 2 eqeq1i ⊢ dom ⁡ ∼ A / ∼ A = A ↔ ⋃ A / ∼ A = A