![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 12480 | . 2 class ↾s | |
2 | vw | . . 3 setvar 𝑤 | |
3 | vx | . . 3 setvar 𝑥 | |
4 | cvv 2751 | . . 3 class V | |
5 | 2 | cv 1362 | . . . 4 class 𝑤 |
6 | cnx 12476 | . . . . . 6 class ndx | |
7 | cbs 12479 | . . . . . 6 class Base | |
8 | 6, 7 | cfv 5230 | . . . . 5 class (Base‘ndx) |
9 | 3 | cv 1362 | . . . . . 6 class 𝑥 |
10 | 5, 7 | cfv 5230 | . . . . . 6 class (Base‘𝑤) |
11 | 9, 10 | cin 3142 | . . . . 5 class (𝑥 ∩ (Base‘𝑤)) |
12 | 8, 11 | cop 3609 | . . . 4 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩ |
13 | csts 12477 | . . . 4 class sSet | |
14 | 5, 12, 13 | co 5890 | . . 3 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩) |
15 | 2, 3, 4, 4, 14 | cmpo 5892 | . 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 12540 ressvalsets 12541 |
Copyright terms: Public domain | W3C validator |