Definition df-struct 17106

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 6505, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17110: 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 4829). This is used critically in strle1 17117, strle2 17118, strle3 17119 and strleun 17116 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17288 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 17289, 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 21355. (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 17105	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1541	. . . . 5 class 𝑥
4		cle 11169	. . . . . 6 class ≤
5		cn 12163	. . . . . . 7 class ℕ
6	5, 5	cxp 5618	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3884	. . . . 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 4263	. . . . . . 7 class ∅
12	11	csn 4557	. . . . . 6 class {∅}
13	10, 12	cdif 3882	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6481	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5620	. . . . 5 class dom 𝑓
16		cfz 13450	. . . . . 6 class ...
17	3, 16	cfv 6487	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3885	. . . 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 5136	. 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  17107  isstruct2  17108