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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 17280 for the altered base set, and resseqnbas 17287 (subrg0 20579, ressplusg 17334, subrg1 20582, ressmulr 17351) 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 17274 | . 2 class ↾s | |
| 2 | vw | . . 3 setvar 𝑤 | |
| 3 | vx | . . 3 setvar 𝑥 | |
| 4 | cvv 3480 | . . 3 class V | |
| 5 | 2 | cv 1539 | . . . . . 6 class 𝑤 |
| 6 | cbs 17247 | . . . . . 6 class Base | |
| 7 | 5, 6 | cfv 6561 | . . . . 5 class (Base‘𝑤) |
| 8 | 3 | cv 1539 | . . . . 5 class 𝑥 |
| 9 | 7, 8 | wss 3951 | . . . 4 wff (Base‘𝑤) ⊆ 𝑥 |
| 10 | cnx 17230 | . . . . . . 7 class ndx | |
| 11 | 10, 6 | cfv 6561 | . . . . . 6 class (Base‘ndx) |
| 12 | 8, 7 | cin 3950 | . . . . . 6 class (𝑥 ∩ (Base‘𝑤)) |
| 13 | 11, 12 | cop 4632 | . . . . 5 class 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉 |
| 14 | csts 17200 | . . . . 5 class sSet | |
| 15 | 5, 13, 14 | co 7431 | . . . 4 class (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉) |
| 16 | 9, 5, 15 | cif 4525 | . . 3 class if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉)) |
| 17 | 2, 3, 4, 4, 16 | cmpo 7433 | . 2 class (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉))) |
| 18 | 1, 17 | wceq 1540 | 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 17276 ressval 17277 |
| Copyright terms: Public domain | W3C validator |