Definition df-sets 17214

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 17281 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 20208, 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 17213	. 2 class sSet
2		vs	. . 3 setvar 𝑠
3		ve	. . 3 setvar 𝑒
4		cvv 3457	. . 3 class V
5	2	cv 1562	. . . . 5 class 𝑠
6	3	cv 1562	. . . . . . . 8 class 𝑒
7	6	csn 4585	. . . . . . 7 class {𝑒}
8	7	cdm 5652	. . . . . 6 class dom {𝑒}
9	4, 8	cdif 3904	. . . . 5 class (V ∖ dom {𝑒})
10	5, 9	cres 5654	. . . 4 class (𝑠 ↾ (V ∖ dom {𝑒}))
11	10, 7	cun 3905	. . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})
12	2, 3, 4, 4, 11	cmpo 7402	. 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
13	1, 12	wceq 1563	1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))

This definition is referenced by:  reldmsets  17215  setsvalg  17216