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Definition df-iress 12469
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, defining a function using the base set and applying that, or explicitly truncating the slot before use.

(Credit for this operator, as well as the 2023 modification for iset.mm, goes to Mario Carneiro.)

(Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 7-Oct-2023.)

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

Detailed syntax breakdown of Definition df-iress
StepHypRef Expression
1 cress 12462 . 2 class s
2 vw . . 3 setvar 𝑤
3 vx . . 3 setvar 𝑥
4 cvv 2737 . . 3 class V
52cv 1352 . . . 4 class 𝑤
6 cnx 12458 . . . . . 6 class ndx
7 cbs 12461 . . . . . 6 class Base
86, 7cfv 5216 . . . . 5 class (Base‘ndx)
93cv 1352 . . . . . 6 class 𝑥
105, 7cfv 5216 . . . . . 6 class (Base‘𝑤)
119, 10cin 3128 . . . . 5 class (𝑥 ∩ (Base‘𝑤))
128, 11cop 3595 . . . 4 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
13 csts 12459 . . . 4 class sSet
145, 12, 13co 5874 . . 3 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
152, 3, 4, 4, 14cmpo 5876 . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
161, 15wceq 1353 1 wff s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
Colors of variables: wff set class
This definition is referenced by:  reldmress  12522  ressvalsets  12523
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