Definition df-struct 17072

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 6508, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17076: 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 4850). This is used critically in strle1 17083, strle2 17084, strle3 17085 and strleun 17082 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17254 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 17255, 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 21321. (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 17071	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1540	. . . . 5 class 𝑥
4		cle 11165	. . . . . 6 class ≤
5		cn 12143	. . . . . . 7 class ℕ
6	5, 5	cxp 5620	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3898	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2113	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1540	. . . . . 6 class 𝑓
11		c0 4283	. . . . . . 7 class ∅
12	11	csn 4578	. . . . . 6 class {∅}
13	10, 12	cdif 3896	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6484	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5622	. . . . 5 class dom 𝑓
16		cfz 13421	. . . . . 6 class ...
17	3, 16	cfv 6490	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3899	. . . 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 5158	. 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  17073  isstruct2  17074