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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 12476 | . 2 class ↾s | |
2 | vw | . . 3 setvar 𝑤 | |
3 | vx | . . 3 setvar 𝑥 | |
4 | cvv 2749 | . . 3 class V | |
5 | 2 | cv 1362 | . . . 4 class 𝑤 |
6 | cnx 12472 | . . . . . 6 class ndx | |
7 | cbs 12475 | . . . . . 6 class Base | |
8 | 6, 7 | cfv 5228 | . . . . 5 class (Base‘ndx) |
9 | 3 | cv 1362 | . . . . . 6 class 𝑥 |
10 | 5, 7 | cfv 5228 | . . . . . 6 class (Base‘𝑤) |
11 | 9, 10 | cin 3140 | . . . . 5 class (𝑥 ∩ (Base‘𝑤)) |
12 | 8, 11 | cop 3607 | . . . 4 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩ |
13 | csts 12473 | . . . 4 class sSet | |
14 | 5, 12, 13 | co 5888 | . . 3 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩) |
15 | 2, 3, 4, 4, 14 | cmpo 5890 | . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)) |
16 | 1, 15 | wceq 1363 | 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 12536 ressvalsets 12537 |
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