Definition df-struct 16946

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 6507, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16950: 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 4845). This is used critically in strle1 16957, strle2 16958, strle3 16959 and strleun 16956 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17144 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 17145, 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 20715. (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 16945	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1540	. . . . 5 class 𝑥
4		cle 11116	. . . . . 6 class ≤
5		cn 12079	. . . . . . 7 class ℕ
6	5, 5	cxp 5623	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3901	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2106	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1540	. . . . . 6 class 𝑓
11		c0 4274	. . . . . . 7 class ∅
12	11	csn 4578	. . . . . 6 class {∅}
13	10, 12	cdif 3899	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6478	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5625	. . . . 5 class dom 𝑓
16		cfz 13345	. . . . . 6 class ...
17	3, 16	cfv 6484	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3902	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1087	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5159	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1541	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  16947  isstruct2  16948