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Definition df-ress 15650
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 15705 for the altered base set, and resslem 15708 (subrg0 18558, ressplusg 15766, subrg1 18561, ressmulr 15777) 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
StepHypRef Expression
1 cress 15644 . 2 class s
2 vw . . 3 setvar 𝑤
3 vx . . 3 setvar 𝑥
4 cvv 3172 . . 3 class V
52cv 1473 . . . . . 6 class 𝑤
6 cbs 15643 . . . . . 6 class Base
75, 6cfv 5789 . . . . 5 class (Base‘𝑤)
83cv 1473 . . . . 5 class 𝑥
97, 8wss 3539 . . . 4 wff (Base‘𝑤) ⊆ 𝑥
10 cnx 15640 . . . . . . 7 class ndx
1110, 6cfv 5789 . . . . . 6 class (Base‘ndx)
128, 7cin 3538 . . . . . 6 class (𝑥 ∩ (Base‘𝑤))
1311, 12cop 4130 . . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14 csts 15641 . . . . 5 class sSet
155, 13, 14co 6526 . . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
169, 5, 15cif 4035 . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
172, 3, 4, 4, 16cmpt2 6528 . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
181, 17wceq 1474 1 wff s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
Colors of variables: wff setvar class
This definition is referenced by:  reldmress  15701  ressval  15702
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