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Definition df-sets 16082
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 16083 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 18699, 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 16073 . 2 class sSet
2 vs . . 3 setvar 𝑠
3 ve . . 3 setvar 𝑒
4 cvv 3402 . . 3 class V
52cv 1636 . . . . 5 class 𝑠
63cv 1636 . . . . . . . 8 class 𝑒
76csn 4381 . . . . . . 7 class {𝑒}
87cdm 5322 . . . . . 6 class dom {𝑒}
94, 8cdif 3777 . . . . 5 class (V ∖ dom {𝑒})
105, 9cres 5324 . . . 4 class (𝑠 ↾ (V ∖ dom {𝑒}))
1110, 7cun 3778 . . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})
122, 3, 4, 4, 11cmpt2 6883 . 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
131, 12wceq 1637 1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
Colors of variables: wff setvar class
This definition is referenced by:  reldmsets  16104  setsvalg  16105
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