Metamath Proof Explorer


Definition df-ptfin

Description: Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010)

Ref Expression
Assertion df-ptfin PtFin = { 𝑥 ∣ ∀ 𝑦 𝑥 { 𝑧𝑥𝑦𝑧 } ∈ Fin }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cptfin PtFin
1 vx 𝑥
2 vy 𝑦
3 1 cv 𝑥
4 3 cuni 𝑥
5 vz 𝑧
6 2 cv 𝑦
7 5 cv 𝑧
8 6 7 wcel 𝑦𝑧
9 8 5 3 crab { 𝑧𝑥𝑦𝑧 }
10 cfn Fin
11 9 10 wcel { 𝑧𝑥𝑦𝑧 } ∈ Fin
12 11 2 4 wral 𝑦 𝑥 { 𝑧𝑥𝑦𝑧 } ∈ Fin
13 12 1 cab { 𝑥 ∣ ∀ 𝑦 𝑥 { 𝑧𝑥𝑦𝑧 } ∈ Fin }
14 0 13 wceq PtFin = { 𝑥 ∣ ∀ 𝑦 𝑥 { 𝑧𝑥𝑦𝑧 } ∈ Fin }