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Definition df-sets 16480
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.)
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 16471 . 2 class sSet
2 vs . . 3 setvar 𝑠
3 ve . . 3 setvar 𝑒
4 cvv 3495 . . 3 class V
52cv 1527 . . . . 5 class 𝑠
63cv 1527 . . . . . . . 8 class 𝑒
76csn 4559 . . . . . . 7 class {𝑒}
87cdm 5549 . . . . . 6 class dom {𝑒}
94, 8cdif 3932 . . . . 5 class (V ∖ dom {𝑒})
105, 9cres 5551 . . . 4 class (𝑠 ↾ (V ∖ dom {𝑒}))
1110, 7cun 3933 . . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})
122, 3, 4, 4, 11cmpo 7147 . 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
131, 12wceq 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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