Definition df-struct 17155

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 6524, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17159: 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 4844). This is used critically in strle1 17166, strle2 17167, strle3 17168 and strleun 17165 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17337 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 17338, 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 21407. (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 17154	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1549	. . . . 5 class 𝑥
4		cle 11203	. . . . . 6 class ≤
5		cn 12196	. . . . . . 7 class ℕ
6	5, 5	cxp 5634	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3894	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2132	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1549	. . . . . 6 class 𝑓
11		c0 4276	. . . . . . 7 class ∅
12	11	csn 4572	. . . . . 6 class {∅}
13	10, 12	cdif 3892	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6500	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5636	. . . . 5 class dom 𝑓
16		cfz 13498	. . . . . 6 class ...
17	3, 16	cfv 6506	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3895	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1095	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5152	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1550	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17156  isstruct2  17157