Metamath Proof Explorer


Theorem dfqs3

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

Ref Expression
Assertion dfqs3 ( 𝐴 / 𝑅 ) = 𝑥𝐴 { [ 𝑥 ] 𝑅 }

Proof

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