MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-sets Structured version   Visualization version   GIF version

Definition df-sets 15643
Description: Set one or more components of a structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 15644 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 18255, 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
StepHypRef Expression
1 csts 15635 . 2 class sSet
2 vs . . 3 setvar 𝑠
3 ve . . 3 setvar 𝑒
4 cvv 3168 . . 3 class V
52cv 1473 . . . . 5 class 𝑠
63cv 1473 . . . . . . . 8 class 𝑒
76csn 4120 . . . . . . 7 class {𝑒}
87cdm 5024 . . . . . 6 class dom {𝑒}
94, 8cdif 3532 . . . . 5 class (V ∖ dom {𝑒})
105, 9cres 5026 . . . 4 class (𝑠 ↾ (V ∖ dom {𝑒}))
1110, 7cun 3533 . . 3 class ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})
122, 3, 4, 4, 11cmpt2 6525 . 2 class (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
131, 12wceq 1474 1 wff sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒}))
Colors of variables: wff setvar class
This definition is referenced by:  reldmsets  15660  setsvalg  15661
  Copyright terms: Public domain W3C validator