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 〉 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( 𝑥 ∩ 𝑧 ) = ( 𝑦 ∩ 𝑧 ) ) } |
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 | ⊢ 𝑥 ⊆ 𝑦 |
7 | 2 | cv | ⊢ 𝑧 |
8 | 4 7 | cin | ⊢ ( 𝑥 ∩ 𝑧 ) |
9 | 5 7 | cin | ⊢ ( 𝑦 ∩ 𝑧 ) |
10 | 8 9 | wceq | ⊢ ( 𝑥 ∩ 𝑧 ) = ( 𝑦 ∩ 𝑧 ) |
11 | 6 10 | wa | ⊢ ( 𝑥 ⊆ 𝑦 ∧ ( 𝑥 ∩ 𝑧 ) = ( 𝑦 ∩ 𝑧 ) ) |
12 | 11 1 2 3 | coprab | ⊢ { 〈 〈 𝑦 , 𝑧 〉 , 𝑥 〉 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( 𝑥 ∩ 𝑧 ) = ( 𝑦 ∩ 𝑧 ) ) } |
13 | 12 | ccnv | ⊢ ◡ { 〈 〈 𝑦 , 𝑧 〉 , 𝑥 〉 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( 𝑥 ∩ 𝑧 ) = ( 𝑦 ∩ 𝑧 ) ) } |
14 | 0 13 | wceq | ⊢ Redunds = ◡ { 〈 〈 𝑦 , 𝑧 〉 , 𝑥 〉 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( 𝑥 ∩ 𝑧 ) = ( 𝑦 ∩ 𝑧 ) ) } |