Metamath Proof Explorer


Theorem dfcoss2

Description: Alternate definition of the class of cosets by R : x and y are cosets by R iff there exists a set u such that both x and y are are elements of the R -coset of u (see also the comment of dfec2 ). R is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018)

Ref Expression
Assertion dfcoss2 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑥 ∈ [ 𝑢 ] 𝑅𝑦 ∈ [ 𝑢 ] 𝑅 ) }

Proof

Step Hyp Ref Expression
1 df-coss 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥𝑢 𝑅 𝑦 ) }
2 elecALTV ( ( 𝑢 ∈ V ∧ 𝑥 ∈ V ) → ( 𝑥 ∈ [ 𝑢 ] 𝑅𝑢 𝑅 𝑥 ) )
3 2 el2v ( 𝑥 ∈ [ 𝑢 ] 𝑅𝑢 𝑅 𝑥 )
4 elecALTV ( ( 𝑢 ∈ V ∧ 𝑦 ∈ V ) → ( 𝑦 ∈ [ 𝑢 ] 𝑅𝑢 𝑅 𝑦 ) )
5 4 el2v ( 𝑦 ∈ [ 𝑢 ] 𝑅𝑢 𝑅 𝑦 )
6 3 5 anbi12i ( ( 𝑥 ∈ [ 𝑢 ] 𝑅𝑦 ∈ [ 𝑢 ] 𝑅 ) ↔ ( 𝑢 𝑅 𝑥𝑢 𝑅 𝑦 ) )
7 6 exbii ( ∃ 𝑢 ( 𝑥 ∈ [ 𝑢 ] 𝑅𝑦 ∈ [ 𝑢 ] 𝑅 ) ↔ ∃ 𝑢 ( 𝑢 𝑅 𝑥𝑢 𝑅 𝑦 ) )
8 7 opabbii { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑥 ∈ [ 𝑢 ] 𝑅𝑦 ∈ [ 𝑢 ] 𝑅 ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑢 𝑅 𝑥𝑢 𝑅 𝑦 ) }
9 1 8 eqtr4i 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ( 𝑥 ∈ [ 𝑢 ] 𝑅𝑦 ∈ [ 𝑢 ] 𝑅 ) }