Definition df-ress 17180

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 17185 for the altered base set, and resseqnbas 17192 (subrg0 20478, ressplusg 17241, subrg1 20481, ressmulr 17258) 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 17179	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 3468	. . 3 class V
5	2	cv 1532	. . . . . 6 class 𝑤
6		cbs 17150	. . . . . 6 class Base
7	5, 6	cfv 6536	. . . . 5 class (Base‘𝑤)
8	3	cv 1532	. . . . 5 class 𝑥
9	7, 8	wss 3943	. . . 4 wff (Base‘𝑤) ⊆ 𝑥
10		cnx 17132	. . . . . . 7 class ndx
11	10, 6	cfv 6536	. . . . . 6 class (Base‘ndx)
12	8, 7	cin 3942	. . . . . 6 class (𝑥 ∩ (Base‘𝑤))
13	11, 12	cop 4629	. . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14		csts 17102	. . . . 5 class sSet
15	5, 13, 14	co 7404	. . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
16	9, 5, 15	cif 4523	. . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
17	2, 3, 4, 4, 16	cmpo 7406	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
18	1, 17	wceq 1533	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  17181  ressval  17182