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Definition df-ress 16072
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 16137 for the altered base set, and resslem 16140 (subrg0 18987, ressplusg 16200, subrg1 18990, ressmulr 16213) 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 16065 . 2 class s
2 vw . . 3 setvar 𝑤
3 vx . . 3 setvar 𝑥
4 cvv 3391 . . 3 class V
52cv 1636 . . . . . 6 class 𝑤
6 cbs 16064 . . . . . 6 class Base
75, 6cfv 6097 . . . . 5 class (Base‘𝑤)
83cv 1636 . . . . 5 class 𝑥
97, 8wss 3769 . . . 4 wff (Base‘𝑤) ⊆ 𝑥
10 cnx 16061 . . . . . . 7 class ndx
1110, 6cfv 6097 . . . . . 6 class (Base‘ndx)
128, 7cin 3768 . . . . . 6 class (𝑥 ∩ (Base‘𝑤))
1311, 12cop 4376 . . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14 csts 16062 . . . . 5 class sSet
155, 13, 14co 6870 . . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
169, 5, 15cif 4279 . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
172, 3, 4, 4, 16cmpt2 6872 . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
181, 17wceq 1637 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  16133  ressval  16134
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