Definition df-ress 16674

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 16738 for the altered base set, and resslem 16741 (subrg0 19761, ressplusg 16796, subrg1 19764, ressmulr 16809) 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 16667	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 3398	. . 3 class V
5	2	cv 1542	. . . . . 6 class 𝑤
6		cbs 16666	. . . . . 6 class Base
7	5, 6	cfv 6358	. . . . 5 class (Base‘𝑤)
8	3	cv 1542	. . . . 5 class 𝑥
9	7, 8	wss 3853	. . . 4 wff (Base‘𝑤) ⊆ 𝑥
10		cnx 16663	. . . . . . 7 class ndx
11	10, 6	cfv 6358	. . . . . 6 class (Base‘ndx)
12	8, 7	cin 3852	. . . . . 6 class (𝑥 ∩ (Base‘𝑤))
13	11, 12	cop 4533	. . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14		csts 16664	. . . . 5 class sSet
15	5, 13, 14	co 7191	. . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
16	9, 5, 15	cif 4425	. . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
17	2, 3, 4, 4, 16	cmpo 7193	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
18	1, 17	wceq 1543	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  16734  ressval  16735