Definition df-ress 17172

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 17177 for the altered base set, and resseqnbas 17183 (subrg0 20529, ressplusg 17225, subrg1 20532, ressmulr 17241) 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 17171	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 3442	. . 3 class V
5	2	cv 1541	. . . . . 6 class 𝑤
6		cbs 17150	. . . . . 6 class Base
7	5, 6	cfv 6502	. . . . 5 class (Base‘𝑤)
8	3	cv 1541	. . . . 5 class 𝑥
9	7, 8	wss 3903	. . . 4 wff (Base‘𝑤) ⊆ 𝑥
10		cnx 17134	. . . . . . 7 class ndx
11	10, 6	cfv 6502	. . . . . 6 class (Base‘ndx)
12	8, 7	cin 3902	. . . . . 6 class (𝑥 ∩ (Base‘𝑤))
13	11, 12	cop 4588	. . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14		csts 17104	. . . . 5 class sSet
15	5, 13, 14	co 7370	. . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
16	9, 5, 15	cif 4481	. . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
17	2, 3, 4, 4, 16	cmpo 7372	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
18	1, 17	wceq 1542	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  17173  ressval  17174