Definition df-struct 17206

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 6554, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17210: 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 4860). This is used critically in strle1 17217, strle2 17218, strle3 17219 and strleun 17216 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17388 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 17389, 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 21515. (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 17205	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1567	. . . . 5 class 𝑥
4		cle 11243	. . . . . 6 class ≤
5		cn 12232	. . . . . . 7 class ℕ
6	5, 5	cxp 5659	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3903	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2141	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1567	. . . . . 6 class 𝑓
11		c0 4285	. . . . . . 7 class ∅
12	11	csn 4588	. . . . . 6 class {∅}
13	10, 12	cdif 3901	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6530	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5661	. . . . 5 class dom 𝑓
16		cfz 13534	. . . . . 6 class ...
17	3, 16	cfv 6536	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3904	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1101	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5172	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1568	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17207  isstruct2  17208