Definition df-struct 17243

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 6555, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17247: 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 4859). This is used critically in strle1 17254, strle2 17255, strle3 17256 and strleun 17253 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17425 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 17426, 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 21600. (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 17242	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1569	. . . . 5 class 𝑥
4		cle 11271	. . . . . 6 class ≤
5		cn 12260	. . . . . . 7 class ℕ
6	5, 5	cxp 5657	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3901	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2145	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1569	. . . . . 6 class 𝑓
11		c0 4282	. . . . . . 7 class ∅
12	11	csn 4587	. . . . . 6 class {∅}
13	10, 12	cdif 3899	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6531	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5659	. . . . 5 class dom 𝑓
16		cfz 13563	. . . . . 6 class ...
17	3, 16	cfv 6537	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3902	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1103	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5171	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1570	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is used by:  brstruct  17244  isstruct2  17245