Description: Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010)
Ref | Expression | ||
---|---|---|---|
Assertion | df-ref | ⊢ Ref = { 〈 𝑥 , 𝑦 〉 ∣ ( ∪ 𝑦 = ∪ 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) } |
Step | Hyp | Ref | Expression |
---|---|---|---|
0 | cref | ⊢ Ref | |
1 | vx | ⊢ 𝑥 | |
2 | vy | ⊢ 𝑦 | |
3 | 2 | cv | ⊢ 𝑦 |
4 | 3 | cuni | ⊢ ∪ 𝑦 |
5 | 1 | cv | ⊢ 𝑥 |
6 | 5 | cuni | ⊢ ∪ 𝑥 |
7 | 4 6 | wceq | ⊢ ∪ 𝑦 = ∪ 𝑥 |
8 | vz | ⊢ 𝑧 | |
9 | vw | ⊢ 𝑤 | |
10 | 8 | cv | ⊢ 𝑧 |
11 | 9 | cv | ⊢ 𝑤 |
12 | 10 11 | wss | ⊢ 𝑧 ⊆ 𝑤 |
13 | 12 9 3 | wrex | ⊢ ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 |
14 | 13 8 5 | wral | ⊢ ∀ 𝑧 ∈ 𝑥 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 |
15 | 7 14 | wa | ⊢ ( ∪ 𝑦 = ∪ 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) |
16 | 15 1 2 | copab | ⊢ { 〈 𝑥 , 𝑦 〉 ∣ ( ∪ 𝑦 = ∪ 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) } |
17 | 0 16 | wceq | ⊢ Ref = { 〈 𝑥 , 𝑦 〉 ∣ ( ∪ 𝑦 = ∪ 𝑥 ∧ ∀ 𝑧 ∈ 𝑥 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) } |