Definition df-struct 16066

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 6115, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16076: 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 4618). This is used critically in strle1 16180, strle2 16181, strle3 16182 and strleun 16179 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 16231 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 16232, which requires that the base set is 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 19962. (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 16060	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1636	. . . . 5 class 𝑥
4		cle 10356	. . . . . 6 class ≤
5		cn 11301	. . . . . . 7 class ℕ
6	5, 5	cxp 5309	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3768	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2156	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1636	. . . . . 6 class 𝑓
11		c0 4116	. . . . . . 7 class ∅
12	11	csn 4370	. . . . . 6 class {∅}
13	10, 12	cdif 3766	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6091	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5311	. . . . 5 class dom 𝑓
16		cfz 12545	. . . . . 6 class ...
17	3, 16	cfv 6097	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3769	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1100	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 4906	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1637	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  16073  isstruct2  16074