Definition df-struct 17068

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 6507, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17072: 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 4849). This is used critically in strle1 17079, strle2 17080, strle3 17081 and strleun 17078 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17250 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 17251, 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 21315. (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 17067	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1540	. . . . 5 class 𝑥
4		cle 11157	. . . . . 6 class ≤
5		cn 12135	. . . . . . 7 class ℕ
6	5, 5	cxp 5619	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3898	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2113	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1540	. . . . . 6 class 𝑓
11		c0 4284	. . . . . . 7 class ∅
12	11	csn 4577	. . . . . 6 class {∅}
13	10, 12	cdif 3896	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6483	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5621	. . . . 5 class dom 𝑓
16		cfz 13417	. . . . . 6 class ...
17	3, 16	cfv 6489	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3899	. . . 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 5157	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1541	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17069  isstruct2  17070