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Definition df-ress 17217
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 17222 for the altered base set, and resseqnbas 17229 (subrg0 20525, ressplusg 17278, subrg1 20528, ressmulr 17295) 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 17216 . 2 class s
2 vw . . 3 setvar 𝑤
3 vx . . 3 setvar 𝑥
4 cvv 3473 . . 3 class V
52cv 1532 . . . . . 6 class 𝑤
6 cbs 17187 . . . . . 6 class Base
75, 6cfv 6553 . . . . 5 class (Base‘𝑤)
83cv 1532 . . . . 5 class 𝑥
97, 8wss 3949 . . . 4 wff (Base‘𝑤) ⊆ 𝑥
10 cnx 17169 . . . . . . 7 class ndx
1110, 6cfv 6553 . . . . . 6 class (Base‘ndx)
128, 7cin 3948 . . . . . 6 class (𝑥 ∩ (Base‘𝑤))
1311, 12cop 4638 . . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14 csts 17139 . . . . 5 class sSet
155, 13, 14co 7426 . . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
169, 5, 15cif 4532 . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
172, 3, 4, 4, 16cmpo 7428 . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
181, 17wceq 1533 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  17218  ressval  17219
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