Definition df-iress 12487

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 12480	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 2751	. . 3 class V
5	2	cv 1362	. . . 4 class 𝑤
6		cnx 12476	. . . . . 6 class ndx
7		cbs 12479	. . . . . 6 class Base
8	6, 7	cfv 5230	. . . . 5 class (Base‘ndx)
9	3	cv 1362	. . . . . 6 class 𝑥
10	5, 7	cfv 5230	. . . . . 6 class (Base‘𝑤)
11	9, 10	cin 3142	. . . . 5 class (𝑥 ∩ (Base‘𝑤))
12	8, 11	cop 3609	. . . 4 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
13		csts 12477	. . . 4 class sSet
14	5, 12, 13	co 5890	. . 3 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
15	2, 3, 4, 4, 14	cmpo 5892	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
16	1, 15	wceq 1363	1 wff ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))

This definition is referenced by:  reldmress  12540  ressvalsets  12541