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Definition df-pmtr 18082
Description: Define a function that generates the transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Assertion
Ref Expression
df-pmtr pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2𝑜} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
Distinct variable group:   𝑝,𝑑,𝑦,𝑧

Detailed syntax breakdown of Definition df-pmtr
StepHypRef Expression
1 cpmtr 18081 . 2 class pmTrsp
2 vd . . 3 setvar 𝑑
3 cvv 3402 . . 3 class V
4 vp . . . 4 setvar 𝑝
5 vy . . . . . . 7 setvar 𝑦
65cv 1636 . . . . . 6 class 𝑦
7 c2o 7799 . . . . . 6 class 2𝑜
8 cen 8198 . . . . . 6 class
96, 7, 8wbr 4855 . . . . 5 wff 𝑦 ≈ 2𝑜
102cv 1636 . . . . . 6 class 𝑑
1110cpw 4362 . . . . 5 class 𝒫 𝑑
129, 5, 11crab 3111 . . . 4 class {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2𝑜}
13 vz . . . . 5 setvar 𝑧
1413, 4wel 2158 . . . . . 6 wff 𝑧𝑝
154cv 1636 . . . . . . . 8 class 𝑝
1613cv 1636 . . . . . . . . 9 class 𝑧
1716csn 4381 . . . . . . . 8 class {𝑧}
1815, 17cdif 3777 . . . . . . 7 class (𝑝 ∖ {𝑧})
1918cuni 4641 . . . . . 6 class (𝑝 ∖ {𝑧})
2014, 19, 16cif 4290 . . . . 5 class if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)
2113, 10, 20cmpt 4934 . . . 4 class (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))
224, 12, 21cmpt 4934 . . 3 class (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2𝑜} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧)))
232, 3, 22cmpt 4934 . 2 class (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2𝑜} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
241, 23wceq 1637 1 wff pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2𝑜} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
Colors of variables: wff setvar class
This definition is referenced by:  pmtrfval  18090
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