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Definition df-sets 15701
 Description: Set one or more components of a structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 15702 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 18313, 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 15693 . 2 class sSet
2 vs . . 3 setvar 𝑠
3 ve . . 3 setvar 𝑒
4 cvv 3173 . . 3 class V
52cv 1474 . . . . 5 class 𝑠
63cv 1474 . . . . . . . 8 class 𝑒
76csn 4125 . . . . . . 7 class {𝑒}
87cdm 5038 . . . . . 6 class dom {𝑒}
94, 8cdif 3537 . . . . 5 class (V ∖ dom {𝑒})
105, 9cres 5040 . . . 4 class (𝑠 ↾ (V ∖ dom {𝑒}))
1110, 7cun 3538 . . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})
122, 3, 4, 4, 11cmpt2 6551 . 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
131, 12wceq 1475 1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
 Colors of variables: wff setvar class This definition is referenced by:  reldmsets  15718  setsvalg  15719
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