Definition df-struct 17149

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 6577, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17153: 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 4902). This is used critically in strle1 17160, strle2 17161, strle3 17162 and strleun 17159 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17350 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 17351, 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 21357. (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 17148	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1533	. . . . 5 class 𝑥
4		cle 11299	. . . . . 6 class ≤
5		cn 12264	. . . . . . 7 class ℕ
6	5, 5	cxp 5680	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3946	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2099	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1533	. . . . . 6 class 𝑓
11		c0 4325	. . . . . . 7 class ∅
12	11	csn 4633	. . . . . 6 class {∅}
13	10, 12	cdif 3944	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6548	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5682	. . . . 5 class dom 𝑓
16		cfz 13538	. . . . . 6 class ...
17	3, 16	cfv 6554	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3947	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1084	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5215	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1534	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17150  isstruct2  17151