Definition df-sets 17262

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 17329 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 20280, 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 17261	. 2 class sSet
2		vs	. . 3 setvar 𝑠
3		ve	. . 3 setvar 𝑒
4		cvv 3453	. . 3 class V
5	2	cv 1569	. . . . 5 class 𝑠
6	3	cv 1569	. . . . . . . 8 class 𝑒
7	6	csn 4587	. . . . . . 7 class {𝑒}
8	7	cdm 5659	. . . . . 6 class dom {𝑒}
9	4, 8	cdif 3899	. . . . 5 class (V ∖ dom {𝑒})
10	5, 9	cres 5661	. . . 4 class (𝑠 ↾ (V ∖ dom {𝑒}))
11	10, 7	cun 3900	. . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})
12	2, 3, 4, 4, 11	cmpo 7419	. 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
13	1, 12	wceq 1570	1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))

This definition is used by:  reldmsets  17263  setsvalg  17264