Definition df-struct 16314

Description: Define a structure with components in 𝑀...𝑁. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems.

As mentioned in the section header, an "extensible structure should be implemented as a function (a set of ordered pairs)". The current definition, however, is less restrictive: it allows for classes which contain the empty set ∅ to be extensible structures. Because of 0nelfun 6243, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16324: 𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅}).

Allowing an extensible structure to contain the empty set ensures that expressions like {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} are structures without asserting or implying that 𝐴, 𝐵, 𝐶 and 𝐷 are sets (if 𝐴 or 𝐵 is a proper class, then ⟨𝐴, 𝐵⟩ = ∅, see opprc 4733). This is used critically in strle1 16421, strle2 16422, strle3 16423 and strleun 16420 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 16472 and theorems like it will have many sethood assumptions, and may not even be usable in the empty context. Instead, the sethood assumption is deferred until it is actually needed, e.g., ipsbase 16473, which requires that the base set be a set but not any of the other components. Usually, a concrete structure like ℂ_fld does not contain the empty set, and therefore is a function, see cnfldfun 20239. (Contributed by Mario Carneiro, 29-Aug-2015.)

Assertion

Ref	Expression
df-struct	⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

Distinct variable group:   𝑥,𝑓

Detailed syntax breakdown of Definition df-struct

Step	Hyp	Ref	Expression
1		cstr 16308	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1521	. . . . 5 class 𝑥
4		cle 10522	. . . . . 6 class ≤
5		cn 11486	. . . . . . 7 class ℕ
6	5, 5	cxp 5441	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3858	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2081	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1521	. . . . . 6 class 𝑓
11		c0 4211	. . . . . . 7 class ∅
12	11	csn 4472	. . . . . 6 class {∅}
13	10, 12	cdif 3856	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6219	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5443	. . . . 5 class dom 𝑓
16		cfz 12742	. . . . . 6 class ...
17	3, 16	cfv 6225	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3859	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1080	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5024	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1522	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  16321  isstruct2  16322