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Definition df-sets 16336
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 16337 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 18953, 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.)
Assertion
Ref Expression
df-sets sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
Distinct variable group:   𝑒,𝑠

Detailed syntax breakdown of Definition df-sets
StepHypRef Expression
1 csts 16327 . 2 class sSet
2 vs . . 3 setvar 𝑠
3 ve . . 3 setvar 𝑒
4 cvv 3409 . . 3 class V
52cv 1506 . . . . 5 class 𝑠
63cv 1506 . . . . . . . 8 class 𝑒
76csn 4435 . . . . . . 7 class {𝑒}
87cdm 5400 . . . . . 6 class dom {𝑒}
94, 8cdif 3822 . . . . 5 class (V ∖ dom {𝑒})
105, 9cres 5402 . . . 4 class (𝑠 ↾ (V ∖ dom {𝑒}))
1110, 7cun 3823 . . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})
122, 3, 4, 4, 11cmpo 6972 . 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
131, 12wceq 1507 1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
Colors of variables: wff setvar class
This definition is referenced by:  reldmsets  16357  setsvalg  16358
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