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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 15977 for the altered base set, and resslem 15980 (subrg0 18835, ressplusg 16040, subrg1 18838, ressmulr 16053) 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 15905 | . 2 class ↾s | |
2 | vw | . . 3 setvar 𝑤 | |
3 | vx | . . 3 setvar 𝑥 | |
4 | cvv 3231 | . . 3 class V | |
5 | 2 | cv 1522 | . . . . . 6 class 𝑤 |
6 | cbs 15904 | . . . . . 6 class Base | |
7 | 5, 6 | cfv 5926 | . . . . 5 class (Base‘𝑤) |
8 | 3 | cv 1522 | . . . . 5 class 𝑥 |
9 | 7, 8 | wss 3607 | . . . 4 wff (Base‘𝑤) ⊆ 𝑥 |
10 | cnx 15901 | . . . . . . 7 class ndx | |
11 | 10, 6 | cfv 5926 | . . . . . 6 class (Base‘ndx) |
12 | 8, 7 | cin 3606 | . . . . . 6 class (𝑥 ∩ (Base‘𝑤)) |
13 | 11, 12 | cop 4216 | . . . . 5 class 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉 |
14 | csts 15902 | . . . . 5 class sSet | |
15 | 5, 13, 14 | co 6690 | . . . 4 class (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉) |
16 | 9, 5, 15 | cif 4119 | . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉)) |
17 | 2, 3, 4, 4, 16 | cmpt2 6692 | . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉))) |
18 | 1, 17 | wceq 1523 | 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 15973 ressval 15974 |
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