Definition df-sets 16482

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 16483 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 19233, 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 16473	. 2 class sSet
2		vs	. . 3 setvar 𝑠
3		ve	. . 3 setvar 𝑒
4		cvv 3441	. . 3 class V
5	2	cv 1537	. . . . 5 class 𝑠
6	3	cv 1537	. . . . . . . 8 class 𝑒
7	6	csn 4525	. . . . . . 7 class {𝑒}
8	7	cdm 5519	. . . . . 6 class dom {𝑒}
9	4, 8	cdif 3878	. . . . 5 class (V ∖ dom {𝑒})
10	5, 9	cres 5521	. . . 4 class (𝑠 ↾ (V ∖ dom {𝑒}))
11	10, 7	cun 3879	. . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})
12	2, 3, 4, 4, 11	cmpo 7137	. 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
13	1, 12	wceq 1538	1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))

This definition is referenced by:  reldmsets  16503  setsvalg  16504