Definition df-iress 13307

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 13300	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 2815	. . 3 class V
5	2	cv 1397	. . . 4 class 𝑤
6		cnx 13296	. . . . . 6 class ndx
7		cbs 13299	. . . . . 6 class Base
8	6, 7	cfv 5357	. . . . 5 class (Base‘ndx)
9	3	cv 1397	. . . . . 6 class 𝑥
10	5, 7	cfv 5357	. . . . . 6 class (Base‘𝑤)
11	9, 10	cin 3213	. . . . 5 class (𝑥 ∩ (Base‘𝑤))
12	8, 11	cop 3697	. . . 4 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
13		csts 13297	. . . 4 class sSet
14	5, 12, 13	co 6058	. . 3 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
15	2, 3, 4, 4, 14	cmpo 6060	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
16	1, 15	wceq 1398	1 wff ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))

This definition is referenced by:  reldmress  13363  ressvalsets  13364