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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 16481 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 19171, 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 16471 | . 2 class sSet | |
2 | vs | . . 3 setvar 𝑠 | |
3 | ve | . . 3 setvar 𝑒 | |
4 | cvv 3495 | . . 3 class V | |
5 | 2 | cv 1527 | . . . . 5 class 𝑠 |
6 | 3 | cv 1527 | . . . . . . . 8 class 𝑒 |
7 | 6 | csn 4559 | . . . . . . 7 class {𝑒} |
8 | 7 | cdm 5549 | . . . . . 6 class dom {𝑒} |
9 | 4, 8 | cdif 3932 | . . . . 5 class (V ∖ dom {𝑒}) |
10 | 5, 9 | cres 5551 | . . . 4 class (𝑠 ↾ (V ∖ dom {𝑒})) |
11 | 10, 7 | cun 3933 | . . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}) |
12 | 2, 3, 4, 4, 11 | cmpo 7147 | . 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) |
13 | 1, 12 | wceq 1528 | 1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) |
Colors of variables: wff setvar class |
This definition is referenced by: reldmsets 16501 setsvalg 16502 |
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