Definition df-struct 17093

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 6518, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17097: 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 4856). This is used critically in strle1 17104, strle2 17105, strle3 17106 and strleun 17103 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17275 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 17276, 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 21310. (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 17092	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1539	. . . . 5 class 𝑥
4		cle 11185	. . . . . 6 class ≤
5		cn 12162	. . . . . . 7 class ℕ
6	5, 5	cxp 5629	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3910	. . . . 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 4292	. . . . . . 7 class ∅
12	11	csn 4585	. . . . . 6 class {∅}
13	10, 12	cdif 3908	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6493	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5631	. . . . 5 class dom 𝑓
16		cfz 13444	. . . . . 6 class ...
17	3, 16	cfv 6499	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3911	. . . 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 5164	. 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  17094  isstruct2  17095