Metamath Proof Explorer


Theorem cossid

Description: Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019)

Ref Expression
Assertion cossid ≀ I = I

Proof

Step Hyp Ref Expression
1 equvinv ( 𝑦 = 𝑧 ↔ ∃ 𝑥 ( 𝑥 = 𝑦𝑥 = 𝑧 ) )
2 ideqg ( 𝑦 ∈ V → ( 𝑥 I 𝑦𝑥 = 𝑦 ) )
3 2 elv ( 𝑥 I 𝑦𝑥 = 𝑦 )
4 ideqg ( 𝑧 ∈ V → ( 𝑥 I 𝑧𝑥 = 𝑧 ) )
5 4 elv ( 𝑥 I 𝑧𝑥 = 𝑧 )
6 3 5 anbi12i ( ( 𝑥 I 𝑦𝑥 I 𝑧 ) ↔ ( 𝑥 = 𝑦𝑥 = 𝑧 ) )
7 6 exbii ( ∃ 𝑥 ( 𝑥 I 𝑦𝑥 I 𝑧 ) ↔ ∃ 𝑥 ( 𝑥 = 𝑦𝑥 = 𝑧 ) )
8 1 7 bitr4i ( 𝑦 = 𝑧 ↔ ∃ 𝑥 ( 𝑥 I 𝑦𝑥 I 𝑧 ) )
9 8 opabbii { ⟨ 𝑦 , 𝑧 ⟩ ∣ 𝑦 = 𝑧 } = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ∃ 𝑥 ( 𝑥 I 𝑦𝑥 I 𝑧 ) }
10 df-id I = { ⟨ 𝑦 , 𝑧 ⟩ ∣ 𝑦 = 𝑧 }
11 df-coss ≀ I = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ∃ 𝑥 ( 𝑥 I 𝑦𝑥 I 𝑧 ) }
12 9 10 11 3eqtr4ri ≀ I = I