Description: Alternate definition of the class of cosets by R (see the comment of df-coss ). (Contributed by Peter Mazsa, 27-Dec-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | dfcoss3 | ⊢ ≀ 𝑅 = ( 𝑅 ∘ ◡ 𝑅 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnvg | ⊢ ( ( 𝑥 ∈ V ∧ 𝑢 ∈ V ) → ( 𝑥 ◡ 𝑅 𝑢 ↔ 𝑢 𝑅 𝑥 ) ) | |
2 | 1 | el2v | ⊢ ( 𝑥 ◡ 𝑅 𝑢 ↔ 𝑢 𝑅 𝑥 ) |
3 | 2 | anbi1i | ⊢ ( ( 𝑥 ◡ 𝑅 𝑢 ∧ 𝑢 𝑅 𝑦 ) ↔ ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑅 𝑦 ) ) |
4 | 3 | exbii | ⊢ ( ∃ 𝑢 ( 𝑥 ◡ 𝑅 𝑢 ∧ 𝑢 𝑅 𝑦 ) ↔ ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑅 𝑦 ) ) |
5 | 4 | opabbii | ⊢ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ( 𝑥 ◡ 𝑅 𝑢 ∧ 𝑢 𝑅 𝑦 ) } = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑅 𝑦 ) } |
6 | df-co | ⊢ ( 𝑅 ∘ ◡ 𝑅 ) = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ( 𝑥 ◡ 𝑅 𝑢 ∧ 𝑢 𝑅 𝑦 ) } | |
7 | df-coss | ⊢ ≀ 𝑅 = { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥 ∧ 𝑢 𝑅 𝑦 ) } | |
8 | 5 6 7 | 3eqtr4ri | ⊢ ≀ 𝑅 = ( 𝑅 ∘ ◡ 𝑅 ) |