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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 )