Definition df-sets 17101

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 17178 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 20029, 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

Step	Hyp	Ref	Expression
1		csts 17100	. 2 class sSet
2		vs	. . 3 setvar 𝑠
3		ve	. . 3 setvar 𝑒
4		cvv 3472	. . 3 class V
5	2	cv 1538	. . . . 5 class 𝑠
6	3	cv 1538	. . . . . . . 8 class 𝑒
7	6	csn 4627	. . . . . . 7 class {𝑒}
8	7	cdm 5675	. . . . . 6 class dom {𝑒}
9	4, 8	cdif 3944	. . . . 5 class (V ∖ dom {𝑒})
10	5, 9	cres 5677	. . . 4 class (𝑠 ↾ (V ∖ dom {𝑒}))
11	10, 7	cun 3945	. . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})
12	2, 3, 4, 4, 11	cmpo 7413	. 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
13	1, 12	wceq 1539	1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))

This definition is referenced by:  reldmsets  17102  setsvalg  17103