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| Mirrors > Home > MPE Home > Th. List > df-sets | Structured version Visualization version GIF version | ||
| Description: Set a component of an extensible structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 17275 adjusts the base set to match its second argument, which has the effect of making subgroups, subspaces, subrings etc. from the original structures. Or df-mgp 20138, which takes a ring and overrides its addition operation with the multiplicative operation, so that we can consider the "multiplicative group" using group and monoid theorems, which expect the operation to be in the +g slot instead of the .r slot. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| Ref | Expression |
|---|---|
| df-sets | ⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | csts 17200 | . 2 class sSet | |
| 2 | vs | . . 3 setvar 𝑠 | |
| 3 | ve | . . 3 setvar 𝑒 | |
| 4 | cvv 3480 | . . 3 class V | |
| 5 | 2 | cv 1539 | . . . . 5 class 𝑠 |
| 6 | 3 | cv 1539 | . . . . . . . 8 class 𝑒 |
| 7 | 6 | csn 4626 | . . . . . . 7 class {𝑒} |
| 8 | 7 | cdm 5685 | . . . . . 6 class dom {𝑒} |
| 9 | 4, 8 | cdif 3948 | . . . . 5 class (V ∖ dom {𝑒}) |
| 10 | 5, 9 | cres 5687 | . . . 4 class (𝑠 ↾ (V ∖ dom {𝑒})) |
| 11 | 10, 7 | cun 3949 | . . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}) |
| 12 | 2, 3, 4, 4, 11 | cmpo 7433 | . 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) |
| 13 | 1, 12 | wceq 1540 | 1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) |
| Colors of variables: wff setvar class |
| This definition is referenced by: reldmsets 17202 setsvalg 17203 |
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