Definition df-iress 12711

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 12704	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 2763	. . 3 class V
5	2	cv 1363	. . . 4 class 𝑤
6		cnx 12700	. . . . . 6 class ndx
7		cbs 12703	. . . . . 6 class Base
8	6, 7	cfv 5259	. . . . 5 class (Base‘ndx)
9	3	cv 1363	. . . . . 6 class 𝑥
10	5, 7	cfv 5259	. . . . . 6 class (Base‘𝑤)
11	9, 10	cin 3156	. . . . 5 class (𝑥 ∩ (Base‘𝑤))
12	8, 11	cop 3626	. . . 4 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
13		csts 12701	. . . 4 class sSet
14	5, 12, 13	co 5925	. . 3 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
15	2, 3, 4, 4, 14	cmpo 5927	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
16	1, 15	wceq 1364	1 wff ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))

This definition is referenced by:  reldmress  12766  ressvalsets  12767