Definition df-struct 17193

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 6539, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17197: 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 17204, strle2 17205, strle3 17206 and strleun 17203 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17375 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 17376, 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 21445. (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 17192	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1560	. . . . 5 class 𝑥
4		cle 11228	. . . . . 6 class ≤
5		cn 12220	. . . . . . 7 class ℕ
6	5, 5	cxp 5646	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3904	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2143	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1560	. . . . . 6 class 𝑓
11		c0 4286	. . . . . . 7 class ∅
12	11	csn 4583	. . . . . 6 class {∅}
13	10, 12	cdif 3902	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6515	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5648	. . . . 5 class dom 𝑓
16		cfz 13522	. . . . . 6 class ...
17	3, 16	cfv 6521	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3905	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1099	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5163	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1561	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17194  isstruct2  17195