Definition df-sets 17103

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 17170 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 20088, 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 17102	. 2 class sSet
2		vs	. . 3 setvar 𝑠
3		ve	. . 3 setvar 𝑒
4		cvv 3442	. . 3 class V
5	2	cv 1541	. . . . 5 class 𝑠
6	3	cv 1541	. . . . . . . 8 class 𝑒
7	6	csn 4582	. . . . . . 7 class {𝑒}
8	7	cdm 5632	. . . . . 6 class dom {𝑒}
9	4, 8	cdif 3900	. . . . 5 class (V ∖ dom {𝑒})
10	5, 9	cres 5634	. . . 4 class (𝑠 ↾ (V ∖ dom {𝑒}))
11	10, 7	cun 3901	. . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})
12	2, 3, 4, 4, 11	cmpo 7370	. 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
13	1, 12	wceq 1542	1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))

This definition is referenced by:  reldmsets  17104  setsvalg  17105