Definition df-iress 13209

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 13202	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 2812	. . 3 class V
5	2	cv 1397	. . . 4 class 𝑤
6		cnx 13198	. . . . . 6 class ndx
7		cbs 13201	. . . . . 6 class Base
8	6, 7	cfv 5351	. . . . 5 class (Base‘ndx)
9	3	cv 1397	. . . . . 6 class 𝑥
10	5, 7	cfv 5351	. . . . . 6 class (Base‘𝑤)
11	9, 10	cin 3209	. . . . 5 class (𝑥 ∩ (Base‘𝑤))
12	8, 11	cop 3691	. . . 4 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
13		csts 13199	. . . 4 class sSet
14	5, 12, 13	co 6049	. . 3 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
15	2, 3, 4, 4, 14	cmpo 6051	. 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  13265  ressvalsets  13266