Description: Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023)
Ref | Expression | ||
---|---|---|---|
Assertion | dfqs3 | ⊢ ( 𝐴 / 𝑅 ) = ∪ 𝑥 ∈ 𝐴 { [ 𝑥 ] 𝑅 } |
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 } |