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Definition df-dilN 34206
Description: Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)
Assertion
Ref Expression
df-dilN Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)}))
Distinct variable group:   𝑘,𝑑,𝑓,𝑥

Detailed syntax breakdown of Definition df-dilN
StepHypRef Expression
1 cdilN 34202 . 2 class Dil
2 vk . . 3 setvar 𝑘
3 cvv 3172 . . 3 class V
4 vd . . . 4 setvar 𝑑
52cv 1473 . . . . 5 class 𝑘
6 catm 33364 . . . . 5 class Atoms
75, 6cfv 5790 . . . 4 class (Atoms‘𝑘)
8 vx . . . . . . . . 9 setvar 𝑥
98cv 1473 . . . . . . . 8 class 𝑥
104cv 1473 . . . . . . . . 9 class 𝑑
11 cwpointsN 34086 . . . . . . . . . 10 class WAtoms
125, 11cfv 5790 . . . . . . . . 9 class (WAtoms‘𝑘)
1310, 12cfv 5790 . . . . . . . 8 class ((WAtoms‘𝑘)‘𝑑)
149, 13wss 3539 . . . . . . 7 wff 𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑)
15 vf . . . . . . . . . 10 setvar 𝑓
1615cv 1473 . . . . . . . . 9 class 𝑓
179, 16cfv 5790 . . . . . . . 8 class (𝑓𝑥)
1817, 9wceq 1474 . . . . . . 7 wff (𝑓𝑥) = 𝑥
1914, 18wi 4 . . . . . 6 wff (𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)
20 cpsubsp 33596 . . . . . . 7 class PSubSp
215, 20cfv 5790 . . . . . 6 class (PSubSp‘𝑘)
2219, 8, 21wral 2895 . . . . 5 wff 𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)
23 cpautN 34087 . . . . . 6 class PAut
245, 23cfv 5790 . . . . 5 class (PAut‘𝑘)
2522, 15, 24crab 2899 . . . 4 class {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)}
264, 7, 25cmpt 4637 . . 3 class (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)})
272, 3, 26cmpt 4637 . 2 class (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)}))
281, 27wceq 1474 1 wff Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓𝑥) = 𝑥)}))
Colors of variables: wff setvar class
This definition is referenced by:  dilfsetN  34253
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