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Definition df-ress 16065
Description: Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range (like Ring), defining a function using the base set and applying that (like TopGrp), or explicitly truncating the slot before use (like MetSp).

(Credit for this operator goes to Mario Carneiro.)

See ressbas 16130 for the altered base set, and resslem 16133 (subrg0 18987, ressplusg 16193, subrg1 18990, ressmulr 16206) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Assertion
Ref Expression
df-ress s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
Distinct variable group:   𝑥,𝑤

Detailed syntax breakdown of Definition df-ress
StepHypRef Expression
1 cress 16058 . 2 class s
2 vw . . 3 setvar 𝑤
3 vx . . 3 setvar 𝑥
4 cvv 3338 . . 3 class V
52cv 1629 . . . . . 6 class 𝑤
6 cbs 16057 . . . . . 6 class Base
75, 6cfv 6047 . . . . 5 class (Base‘𝑤)
83cv 1629 . . . . 5 class 𝑥
97, 8wss 3713 . . . 4 wff (Base‘𝑤) ⊆ 𝑥
10 cnx 16054 . . . . . . 7 class ndx
1110, 6cfv 6047 . . . . . 6 class (Base‘ndx)
128, 7cin 3712 . . . . . 6 class (𝑥 ∩ (Base‘𝑤))
1311, 12cop 4325 . . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14 csts 16055 . . . . 5 class sSet
155, 13, 14co 6811 . . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
169, 5, 15cif 4228 . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
172, 3, 4, 4, 16cmpt2 6813 . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
181, 17wceq 1630 1 wff s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
Colors of variables: wff setvar class
This definition is referenced by:  reldmress  16126  ressval  16127
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