Definition df-struct 16857

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 6459, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16861: 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 4828). This is used critically in strle1 16868, strle2 16869, strle3 16870 and strleun 16867 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17055 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 17056, 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 20618. (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 16856	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1538	. . . . 5 class 𝑥
4		cle 11019	. . . . . 6 class ≤
5		cn 11982	. . . . . . 7 class ℕ
6	5, 5	cxp 5588	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3887	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2107	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1538	. . . . . 6 class 𝑓
11		c0 4257	. . . . . . 7 class ∅
12	11	csn 4562	. . . . . 6 class {∅}
13	10, 12	cdif 3885	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6431	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5590	. . . . 5 class dom 𝑓
16		cfz 13248	. . . . . 6 class ...
17	3, 16	cfv 6437	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3888	. . . 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 5137	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1539	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  16858  isstruct2  16859