Definition df-iress 12884

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

Step	Hyp	Ref	Expression
1		cress 12877	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 2773	. . 3 class V
5	2	cv 1372	. . . 4 class 𝑤
6		cnx 12873	. . . . . 6 class ndx
7		cbs 12876	. . . . . 6 class Base
8	6, 7	cfv 5276	. . . . 5 class (Base‘ndx)
9	3	cv 1372	. . . . . 6 class 𝑥
10	5, 7	cfv 5276	. . . . . 6 class (Base‘𝑤)
11	9, 10	cin 3166	. . . . 5 class (𝑥 ∩ (Base‘𝑤))
12	8, 11	cop 3637	. . . 4 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
13		csts 12874	. . . 4 class sSet
14	5, 12, 13	co 5951	. . 3 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
15	2, 3, 4, 4, 14	cmpo 5953	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
16	1, 15	wceq 1373	1 wff ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))

This definition is referenced by:  reldmress  12939  ressvalsets  12940