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Definition df-slot 16477
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 17546) or a group (df-grp 18046)).

Note that Slot 𝐴 is implemented as "evaluation at 𝐴". That is, (Slot 𝐴𝑆) is defined to be (𝑆𝐴), 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 , can be used to extract the number 𝐴 from a slot, since (Slot 𝐴‘ndx) = 𝐴 (see ndxarg 16498). 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 ↦ (𝑓𝑥))) 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 6677). 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 ↦ (𝑥𝐴))
Distinct variable group:   𝑥,𝐴

Detailed syntax breakdown of Definition df-slot
StepHypRef Expression
1 cA . . 3 class 𝐴
21cslot 16472 . 2 class Slot 𝐴
3 vx . . 3 setvar 𝑥
4 cvv 3495 . . 3 class V
53cv 1527 . . . 4 class 𝑥
61, 5cfv 6349 . . 3 class (𝑥𝐴)
73, 4, 6cmpt 5138 . 2 class (𝑥 ∈ V ↦ (𝑥𝐴))
82, 7wceq 1528 1 wff Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥𝐴))
Colors of variables: wff setvar class
This definition is referenced by:  sloteq  16478  slotfn  16491  strfvnd  16492  bj-evaleq  34258  bj-evalfun  34259  bj-evalfn  34260  bj-evalval  34261
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