Definition df-ress 11749

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 11742	. 2 class ↾s
2		vw	. . 3 setvar 𝑤
3		vx	. . 3 setvar 𝑥
4		cvv 2641	. . 3 class V
5	2	cv 1298	. . . . . 6 class 𝑤
6		cbs 11741	. . . . . 6 class Base
7	5, 6	cfv 5059	. . . . 5 class (Base‘𝑤)
8	3	cv 1298	. . . . 5 class 𝑥
9	7, 8	wss 3021	. . . 4 wff (Base‘𝑤) ⊆ 𝑥
10		cnx 11738	. . . . . . 7 class ndx
11	10, 6	cfv 5059	. . . . . 6 class (Base‘ndx)
12	8, 7	cin 3020	. . . . . 6 class (𝑥 ∩ (Base‘𝑤))
13	11, 12	cop 3477	. . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14		csts 11739	. . . . 5 class sSet
15	5, 13, 14	co 5706	. . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
16	9, 5, 15	cif 3421	. . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
17	2, 3, 4, 4, 16	cmpo 5708	. 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
18	1, 17	wceq 1299	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  11799  ressid2  11800  ressval2  11801