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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 16544 for the altered base set, and resslem 16547 (subrg0 19473, ressplusg 16602, subrg1 19476, ressmulr 16615) 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 16474 | . 2 class ↾s | |
2 | vw | . . 3 setvar 𝑤 | |
3 | vx | . . 3 setvar 𝑥 | |
4 | cvv 3495 | . . 3 class V | |
5 | 2 | cv 1527 | . . . . . 6 class 𝑤 |
6 | cbs 16473 | . . . . . 6 class Base | |
7 | 5, 6 | cfv 6349 | . . . . 5 class (Base‘𝑤) |
8 | 3 | cv 1527 | . . . . 5 class 𝑥 |
9 | 7, 8 | wss 3935 | . . . 4 wff (Base‘𝑤) ⊆ 𝑥 |
10 | cnx 16470 | . . . . . . 7 class ndx | |
11 | 10, 6 | cfv 6349 | . . . . . 6 class (Base‘ndx) |
12 | 8, 7 | cin 3934 | . . . . . 6 class (𝑥 ∩ (Base‘𝑤)) |
13 | 11, 12 | cop 4565 | . . . . 5 class 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉 |
14 | csts 16471 | . . . . 5 class sSet | |
15 | 5, 13, 14 | co 7145 | . . . 4 class (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉) |
16 | 9, 5, 15 | cif 4465 | . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉)) |
17 | 2, 3, 4, 4, 16 | cmpo 7147 | . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉))) |
18 | 1, 17 | wceq 1528 | 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 16540 ressval 16541 |
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