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Mirrors > Home > MPE Home > Th. List > df-ress | Structured version Visualization version 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
(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 16410 for the altered base set, and resslem 16413 (subrg0 19265, ressplusg 16468, subrg1 19268, ressmulr 16481) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.) |
Ref | Expression |
---|---|
df-ress | ⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cress 16340 | . 2 class ↾s | |
2 | vw | . . 3 setvar 𝑤 | |
3 | vx | . . 3 setvar 𝑥 | |
4 | cvv 3415 | . . 3 class V | |
5 | 2 | cv 1506 | . . . . . 6 class 𝑤 |
6 | cbs 16339 | . . . . . 6 class Base | |
7 | 5, 6 | cfv 6188 | . . . . 5 class (Base‘𝑤) |
8 | 3 | cv 1506 | . . . . 5 class 𝑥 |
9 | 7, 8 | wss 3829 | . . . 4 wff (Base‘𝑤) ⊆ 𝑥 |
10 | cnx 16336 | . . . . . . 7 class ndx | |
11 | 10, 6 | cfv 6188 | . . . . . 6 class (Base‘ndx) |
12 | 8, 7 | cin 3828 | . . . . . 6 class (𝑥 ∩ (Base‘𝑤)) |
13 | 11, 12 | cop 4447 | . . . . 5 class 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉 |
14 | csts 16337 | . . . . 5 class sSet | |
15 | 5, 13, 14 | co 6976 | . . . 4 class (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉) |
16 | 9, 5, 15 | cif 4350 | . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉)) |
17 | 2, 3, 4, 4, 16 | cmpo 6978 | . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉))) |
18 | 1, 17 | wceq 1507 | 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 16406 ressval 16407 |
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