Definition df-ress 17290

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 17295 for the altered base set, and resseqnbas 17301 (subrg0 20663, ressplusg 17343, subrg1 20666, ressmulr 17359) 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 17289	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 3453	. . 3 class V
5	2	cv 1567	. . . . . 6 class 𝑤
6		cbs 17268	. . . . . 6 class Base
7	5, 6	cfv 6536	. . . . 5 class (Base‘𝑤)
8	3	cv 1567	. . . . 5 class 𝑥
9	7, 8	wss 3904	. . . 4 wff (Base‘𝑤) ⊆ 𝑥
10		cnx 17252	. . . . . . 7 class ndx
11	10, 6	cfv 6536	. . . . . 6 class (Base‘ndx)
12	8, 7	cin 3903	. . . . . 6 class (𝑥 ∩ (Base‘𝑤))
13	11, 12	cop 4594	. . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14		csts 17222	. . . . 5 class sSet
15	5, 13, 14	co 7410	. . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
16	9, 5, 15	cif 4486	. . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
17	2, 3, 4, 4, 16	cmpo 7412	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
18	1, 17	wceq 1568	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  17291  ressval  17292