Definition df-slot 16479

Description: Define the slot extractor for extensible structures. The class Slot 𝐴 is a function whose argument can be any set, although it is meaningful only if that set is a member of an extensible structure (such as a partially ordered set (df-poset 17548) or a group (df-grp 18098)).

Note that Slot 𝐴 is implemented as "evaluation at 𝐴". That is, (Slot 𝐴‘𝑆) is defined to be (𝑆‘𝐴), where 𝐴 will typically be a small nonzero natural number. Each extensible structure 𝑆 is a function defined on specific natural number "slots", and this function extracts the value at a particular slot.

The special "structure" ndx, defined as the identity function restricted to ℕ, can be used to extract the number 𝐴 from a slot, since (Slot 𝐴‘ndx) = 𝐴 (see ndxarg 16500). This is typically used to refer to the number of a slot when defining structures without having to expose the detail of what that number is (for instance, we use the expression (Base‘ndx) in theorems and proofs instead of its value 1).

The class Slot cannot be defined as (𝑥 ∈ V ↦ (𝑓 ∈ V ↦ (𝑓‘𝑥))) because each Slot 𝐴 is a function on the proper class V so is itself a proper class, and the values of functions are sets (fvex 6658). It is necessary to allow proper classes as values of Slot 𝐴 since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.)

Assertion

Ref	Expression
df-slot	⊢ Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴))

Distinct variable group:   𝑥,𝐴

Detailed syntax breakdown of Definition df-slot

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	cslot 16474	. 2 class Slot 𝐴
3		vx	. . 3 setvar 𝑥
4		cvv 3441	. . 3 class V
5	3	cv 1537	. . . 4 class 𝑥
6	1, 5	cfv 6324	. . 3 class (𝑥‘𝐴)
7	3, 4, 6	cmpt 5110	. 2 class (𝑥 ∈ V ↦ (𝑥‘𝐴))
8	2, 7	wceq 1538	1 wff Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴))

This definition is referenced by:  sloteq  16480  slotfn  16493  strfvnd  16494  bj-evaleq  34487  bj-evalfun  34488  bj-evalfn  34489  bj-evalval  34490