Definition df-struct 17286

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 6545, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17290: 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 4855). This is used critically in strle1 17297, strle2 17298, strle3 17299 and strleun 17296 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17468 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 17469, 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 21653. (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 17285	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1569	. . . . 5 class 𝑥
4		cle 11315	. . . . . 6 class ≤
5		cn 12304	. . . . . . 7 class ℕ
6	5, 5	cxp 5645	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3897	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2145	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1569	. . . . . 6 class 𝑓
11		c0 4278	. . . . . . 7 class ∅
12	11	csn 4583	. . . . . 6 class {∅}
13	10, 12	cdif 3895	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6521	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5647	. . . . 5 class dom 𝑓
16		cfz 13608	. . . . . 6 class ...
17	3, 16	cfv 6527	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3898	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1103	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5166	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1570	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is used by:  brstruct  17287  isstruct2  17288