Definition df-struct 16776

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 6436, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16780: 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 4824). This is used critically in strle1 16787, strle2 16788, strle3 16789 and strleun 16786 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 16971 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 16972, 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 20522. (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 16775	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1538	. . . . 5 class 𝑥
4		cle 10941	. . . . . 6 class ≤
5		cn 11903	. . . . . . 7 class ℕ
6	5, 5	cxp 5578	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3882	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2108	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1538	. . . . . 6 class 𝑓
11		c0 4253	. . . . . . 7 class ∅
12	11	csn 4558	. . . . . 6 class {∅}
13	10, 12	cdif 3880	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6412	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5580	. . . . 5 class dom 𝑓
16		cfz 13168	. . . . . 6 class ...
17	3, 16	cfv 6418	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3883	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1085	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5132	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1539	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  16777  isstruct2  16778