Definition df-struct 17189

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 6595, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17193: 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 4920). This is used critically in strle1 17200, strle2 17201, strle3 17202 and strleun 17199 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17390 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 17391, 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 21396. (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 17188	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1536	. . . . 5 class 𝑥
4		cle 11321	. . . . . 6 class ≤
5		cn 12289	. . . . . . 7 class ℕ
6	5, 5	cxp 5697	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3969	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2103	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1536	. . . . . 6 class 𝑓
11		c0 4347	. . . . . . 7 class ∅
12	11	csn 4648	. . . . . 6 class {∅}
13	10, 12	cdif 3967	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6566	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5699	. . . . 5 class dom 𝑓
16		cfz 13563	. . . . . 6 class ...
17	3, 16	cfv 6572	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3970	. . . 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 5231	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1537	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17190  isstruct2  17191