Definition df-struct 17115

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 6566, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17119: 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 4892). This is used critically in strle1 17126, strle2 17127, strle3 17128 and strleun 17125 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17316 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 17317, 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 21297. (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 17114	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1532	. . . . 5 class 𝑥
4		cle 11279	. . . . . 6 class ≤
5		cn 12242	. . . . . . 7 class ℕ
6	5, 5	cxp 5670	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3938	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2098	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1532	. . . . . 6 class 𝑓
11		c0 4318	. . . . . . 7 class ∅
12	11	csn 4624	. . . . . 6 class {∅}
13	10, 12	cdif 3936	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6537	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5672	. . . . 5 class dom 𝑓
16		cfz 13516	. . . . . 6 class ...
17	3, 16	cfv 6543	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3939	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1084	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5205	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1533	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17116  isstruct2  17117