Metamath Proof Explorer


Definition df-redunds

Description: Define the class of all redundant sets x with respect to y in z . For sets, binary relation on the class of all redundant sets ( brredunds ) is equivalent to satisfying the redundancy predicate ( df-redund ). (Contributed by Peter Mazsa, 23-Oct-2022)

Ref Expression
Assertion df-redunds Redunds = { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ ( 𝑥𝑦 ∧ ( 𝑥𝑧 ) = ( 𝑦𝑧 ) ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 credunds Redunds
1 vy 𝑦
2 vz 𝑧
3 vx 𝑥
4 3 cv 𝑥
5 1 cv 𝑦
6 4 5 wss 𝑥𝑦
7 2 cv 𝑧
8 4 7 cin ( 𝑥𝑧 )
9 5 7 cin ( 𝑦𝑧 )
10 8 9 wceq ( 𝑥𝑧 ) = ( 𝑦𝑧 )
11 6 10 wa ( 𝑥𝑦 ∧ ( 𝑥𝑧 ) = ( 𝑦𝑧 ) )
12 11 1 2 3 coprab { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ ( 𝑥𝑦 ∧ ( 𝑥𝑧 ) = ( 𝑦𝑧 ) ) }
13 12 ccnv { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ ( 𝑥𝑦 ∧ ( 𝑥𝑧 ) = ( 𝑦𝑧 ) ) }
14 0 13 wceq Redunds = { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ ( 𝑥𝑦 ∧ ( 𝑥𝑧 ) = ( 𝑦𝑧 ) ) }