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Definition df-ress 16347
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 16410 for the altered base set, and resslem 16413 (subrg0 19265, ressplusg 16468, subrg1 19268, ressmulr 16481) 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 16340 . 2 class s
2 vw . . 3 setvar 𝑤
3 vx . . 3 setvar 𝑥
4 cvv 3415 . . 3 class V
52cv 1506 . . . . . 6 class 𝑤
6 cbs 16339 . . . . . 6 class Base
75, 6cfv 6188 . . . . 5 class (Base‘𝑤)
83cv 1506 . . . . 5 class 𝑥
97, 8wss 3829 . . . 4 wff (Base‘𝑤) ⊆ 𝑥
10 cnx 16336 . . . . . . 7 class ndx
1110, 6cfv 6188 . . . . . 6 class (Base‘ndx)
128, 7cin 3828 . . . . . 6 class (𝑥 ∩ (Base‘𝑤))
1311, 12cop 4447 . . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14 csts 16337 . . . . 5 class sSet
155, 13, 14co 6976 . . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
169, 5, 15cif 4350 . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
172, 3, 4, 4, 16cmpo 6978 . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
181, 17wceq 1507 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  16406  ressval  16407
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