Metamath Proof Explorer


Definition df-ref

Description: Define the refinement relation. (Contributed by Jeff Hankins, 18-Jan-2010)

Ref Expression
Assertion df-ref Ref = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀ 𝑧𝑥𝑤𝑦 𝑧𝑤 ) }

Detailed syntax breakdown

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 𝑧𝑤
13 12 9 3 wrex 𝑤𝑦 𝑧𝑤
14 13 8 5 wral 𝑧𝑥𝑤𝑦 𝑧𝑤
15 7 14 wa ( 𝑦 = 𝑥 ∧ ∀ 𝑧𝑥𝑤𝑦 𝑧𝑤 )
16 15 1 2 copab { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀ 𝑧𝑥𝑤𝑦 𝑧𝑤 ) }
17 0 16 wceq Ref = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑦 = 𝑥 ∧ ∀ 𝑧𝑥𝑤𝑦 𝑧𝑤 ) }