Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-ress Structured version   Visualization version   GIF version

Definition df-ress 15702
 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 15757 for the altered base set, and resslem 15760 (subrg0 18610, ressplusg 15818, subrg1 18613, ressmulr 15829) 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 15696 . 2 class s
2 vw . . 3 setvar 𝑤
3 vx . . 3 setvar 𝑥
4 cvv 3173 . . 3 class V
52cv 1474 . . . . . 6 class 𝑤
6 cbs 15695 . . . . . 6 class Base
75, 6cfv 5804 . . . . 5 class (Base‘𝑤)
83cv 1474 . . . . 5 class 𝑥
97, 8wss 3540 . . . 4 wff (Base‘𝑤) ⊆ 𝑥
10 cnx 15692 . . . . . . 7 class ndx
1110, 6cfv 5804 . . . . . 6 class (Base‘ndx)
128, 7cin 3539 . . . . . 6 class (𝑥 ∩ (Base‘𝑤))
1311, 12cop 4131 . . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩
14 csts 15693 . . . . 5 class sSet
155, 13, 14co 6549 . . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)
169, 5, 15cif 4036 . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))
172, 3, 4, 4, 16cmpt2 6551 . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)))
181, 17wceq 1475 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  15753  ressval  15754
 Copyright terms: Public domain W3C validator