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Mirrors > Home > ILE Home > Th. List > df-iress | GIF version |
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, as well as the 2023 modification for iset.mm, goes to Mario Carneiro.) (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 7-Oct-2023.) |
Ref | Expression |
---|---|
df-iress | ⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cress 12462 | . 2 class ↾s | |
2 | vw | . . 3 setvar 𝑤 | |
3 | vx | . . 3 setvar 𝑥 | |
4 | cvv 2737 | . . 3 class V | |
5 | 2 | cv 1352 | . . . 4 class 𝑤 |
6 | cnx 12458 | . . . . . 6 class ndx | |
7 | cbs 12461 | . . . . . 6 class Base | |
8 | 6, 7 | cfv 5216 | . . . . 5 class (Base‘ndx) |
9 | 3 | cv 1352 | . . . . . 6 class 𝑥 |
10 | 5, 7 | cfv 5216 | . . . . . 6 class (Base‘𝑤) |
11 | 9, 10 | cin 3128 | . . . . 5 class (𝑥 ∩ (Base‘𝑤)) |
12 | 8, 11 | cop 3595 | . . . 4 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩ |
13 | csts 12459 | . . . 4 class sSet | |
14 | 5, 12, 13 | co 5874 | . . 3 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩) |
15 | 2, 3, 4, 4, 14 | cmpo 5876 | . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)) |
16 | 1, 15 | wceq 1353 | 1 wff ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)) |
Colors of variables: wff set class |
This definition is referenced by: reldmress 12522 ressvalsets 12523 |
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