Theorem dfqs3

Description: Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023)

	Ref	Expression
Assertion	dfqs3	⊢ ( 𝐴 / 𝑅 ) = ∪ 𝑥 ∈ 𝐴 { [ 𝑥 ] 𝑅 }

Proof

Step	Hyp	Ref	Expression
1		df-qs	⊢ ( 𝐴 / 𝑅 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = [ 𝑥 ] 𝑅 }
2		iunsn	⊢ ∪ 𝑥 ∈ 𝐴 { [ 𝑥 ] 𝑅 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = [ 𝑥 ] 𝑅 }
3	1 2	eqtr4i	⊢ ( 𝐴 / 𝑅 ) = ∪ 𝑥 ∈ 𝐴 { [ 𝑥 ] 𝑅 }