Definition df-struct 17114

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 6514, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17118: 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 4840). This is used critically in strle1 17125, strle2 17126, strle3 17127 and strleun 17124 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17296 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 17297, 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 21363. (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 17113	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1541	. . . . 5 class 𝑥
4		cle 11177	. . . . . 6 class ≤
5		cn 12171	. . . . . . 7 class ℕ
6	5, 5	cxp 5626	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3889	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2114	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1541	. . . . . 6 class 𝑓
11		c0 4274	. . . . . . 7 class ∅
12	11	csn 4568	. . . . . 6 class {∅}
13	10, 12	cdif 3887	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6490	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5628	. . . . 5 class dom 𝑓
16		cfz 13458	. . . . . 6 class ...
17	3, 16	cfv 6496	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3890	. . . 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 5148	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1542	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17115  isstruct2  17116