Definition df-ress 17160

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 (like Ring), defining a function using the base set and applying that (like TopGrp), or explicitly truncating the slot before use (like MetSp).

(Credit for this operator goes to Mario Carneiro.)

See ressbas 17165 for the altered base set, and resseqnbas 17171 (subrg0 20482, ressplusg 17213, subrg1 20485, ressmulr 17229) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Assertion

Ref	Expression
df-ress	⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))

Distinct variable group:   𝑥,𝑤

Detailed syntax breakdown of Definition df-ress

Step	Hyp	Ref	Expression
1		cress 17159	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 3438	. . 3 class V
5	2	cv 1539	. . . . . 6 class 𝑤
6		cbs 17138	. . . . . 6 class Base
7	5, 6	cfv 6486	. . . . 5 class (Base‘𝑤)
8	3	cv 1539	. . . . 5 class 𝑥
9	7, 8	wss 3905	. . . 4 wff (Base‘𝑤) ⊆ 𝑥
10		cnx 17122	. . . . . . 7 class ndx
11	10, 6	cfv 6486	. . . . . 6 class (Base‘ndx)
12	8, 7	cin 3904	. . . . . 6 class (𝑥 ∩ (Base‘𝑤))
13	11, 12	cop 4585	. . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14		csts 17092	. . . . 5 class sSet
15	5, 13, 14	co 7353	. . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
16	9, 5, 15	cif 4478	. . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
17	2, 3, 4, 4, 16	cmpo 7355	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
18	1, 17	wceq 1540	1 wff ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))

This definition is referenced by:  reldmress  17161  ressval  17162