Definition df-struct 17089

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 6560, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17093: 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 4891). This is used critically in strle1 17100, strle2 17101, strle3 17102 and strleun 17099 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17290 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 17291, 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 21254. (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 17088	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1532	. . . . 5 class 𝑥
4		cle 11253	. . . . . 6 class ≤
5		cn 12216	. . . . . . 7 class ℕ
6	5, 5	cxp 5667	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3942	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2098	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1532	. . . . . 6 class 𝑓
11		c0 4317	. . . . . . 7 class ∅
12	11	csn 4623	. . . . . 6 class {∅}
13	10, 12	cdif 3940	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6531	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5669	. . . . 5 class dom 𝑓
16		cfz 13490	. . . . . 6 class ...
17	3, 16	cfv 6537	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3943	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1084	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5203	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1533	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17090  isstruct2  17091