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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 17222 for the altered base set, and resseqnbas 17229 (subrg0 20525, ressplusg 17278, subrg1 20528, ressmulr 17295) 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 17216 | . 2 class ↾s | |
2 | vw | . . 3 setvar 𝑤 | |
3 | vx | . . 3 setvar 𝑥 | |
4 | cvv 3473 | . . 3 class V | |
5 | 2 | cv 1532 | . . . . . 6 class 𝑤 |
6 | cbs 17187 | . . . . . 6 class Base | |
7 | 5, 6 | cfv 6553 | . . . . 5 class (Base‘𝑤) |
8 | 3 | cv 1532 | . . . . 5 class 𝑥 |
9 | 7, 8 | wss 3949 | . . . 4 wff (Base‘𝑤) ⊆ 𝑥 |
10 | cnx 17169 | . . . . . . 7 class ndx | |
11 | 10, 6 | cfv 6553 | . . . . . 6 class (Base‘ndx) |
12 | 8, 7 | cin 3948 | . . . . . 6 class (𝑥 ∩ (Base‘𝑤)) |
13 | 11, 12 | cop 4638 | . . . . 5 class ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩ |
14 | csts 17139 | . . . . 5 class sSet | |
15 | 5, 13, 14 | co 7426 | . . . 4 class (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩) |
16 | 9, 5, 15 | cif 4532 | . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩)) |
17 | 2, 3, 4, 4, 16 | cmpo 7428 | . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑥 ∩ (Base‘𝑤))⟩))) |
18 | 1, 17 | wceq 1533 | 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 17218 ressval 17219 |
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