Definition df-struct 17123

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 6542, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17127: 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 4868). This is used critically in strle1 17134, strle2 17135, strle3 17136 and strleun 17133 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17305 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 17306, 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 21284. (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 17122	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1539	. . . . 5 class 𝑥
4		cle 11227	. . . . . 6 class ≤
5		cn 12197	. . . . . . 7 class ℕ
6	5, 5	cxp 5644	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3921	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2109	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1539	. . . . . 6 class 𝑓
11		c0 4304	. . . . . . 7 class ∅
12	11	csn 4597	. . . . . 6 class {∅}
13	10, 12	cdif 3919	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6513	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5646	. . . . 5 class dom 𝑓
16		cfz 13481	. . . . . 6 class ...
17	3, 16	cfv 6519	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3922	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1086	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5177	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1540	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17124  isstruct2  17125