Definition df-ress 12006

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, defining a function using the base set and applying that, or explicitly truncating the slot before use.

(Credit for this operator goes to Mario Carneiro.)

(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 11999	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 2689	. . 3 class V
5	2	cv 1331	. . . . . 6 class 𝑤
6		cbs 11998	. . . . . 6 class Base
7	5, 6	cfv 5131	. . . . 5 class (Base‘𝑤)
8	3	cv 1331	. . . . 5 class 𝑥
9	7, 8	wss 3076	. . . 4 wff (Base‘𝑤) ⊆ 𝑥
10		cnx 11995	. . . . . . 7 class ndx
11	10, 6	cfv 5131	. . . . . 6 class (Base‘ndx)
12	8, 7	cin 3075	. . . . . 6 class (𝑥 ∩ (Base‘𝑤))
13	11, 12	cop 3535	. . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14		csts 11996	. . . . 5 class sSet
15	5, 13, 14	co 5782	. . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
16	9, 5, 15	cif 3479	. . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
17	2, 3, 4, 4, 16	cmpo 5784	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
18	1, 17	wceq 1332	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  12056  ressid2  12057  ressval2  12058