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Metamath Proof Explorer

Theorem dmqs1cosscnvepreseq

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 dmqs1cosscnvepreseq ⊢ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A ↔ ⋃ A / ∼ A = A

Proof

Step Hyp Ref Expression
1 df-coels ⊢ ∼ A = ≀ E -1 ↾ A
2 1 dmqseqeq1i ⊢ dom ⁡ ∼ A / ∼ A = A ↔ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A
3 dmqscoelseq ⊢ dom ⁡ ∼ A / ∼ A = A ↔ ⋃ A / ∼ A = A
4 2 3 bitr3i ⊢ dom ⁡ ≀ E -1 ↾ A / ≀ E -1 ↾ A = A ↔ ⋃ A / ∼ A = A