Description: Alternate definition of the class of cosets by R (see the comment of df-coss ). (Contributed by Peter Mazsa, 12-Jul-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | dfcoss4 | ⊢ ≀ 𝑅 = ran ( 𝑅 ⋉ 𝑅 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coss | ⊢ ≀ 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑅 𝑦 ) } | |
2 | rnxrn | ⊢ ran ( 𝑅 ⋉ 𝑅 ) = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑅 𝑦 ) } | |
3 | 1 2 | eqtr4i | ⊢ ≀ 𝑅 = ran ( 𝑅 ⋉ 𝑅 ) |