Definition df-struct 17058

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 6500, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17062: 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 4847). This is used critically in strle1 17069, strle2 17070, strle3 17071 and strleun 17068 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17240 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 17241, 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 21275. (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 17057	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1539	. . . . 5 class 𝑥
4		cle 11150	. . . . . 6 class ≤
5		cn 12128	. . . . . . 7 class ℕ
6	5, 5	cxp 5617	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3902	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2109	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1539	. . . . . 6 class 𝑓
11		c0 4284	. . . . . . 7 class ∅
12	11	csn 4577	. . . . . 6 class {∅}
13	10, 12	cdif 3900	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6476	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5619	. . . . 5 class dom 𝑓
16		cfz 13410	. . . . . 6 class ...
17	3, 16	cfv 6482	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3903	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1086	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5154	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1540	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17059  isstruct2  17060