Definition df-struct 17088

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 6520, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17092: 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 4854). This is used critically in strle1 17099, strle2 17100, strle3 17101 and strleun 17098 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17270 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 17271, 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 21340. (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 17087	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1541	. . . . 5 class 𝑥
4		cle 11181	. . . . . 6 class ≤
5		cn 12159	. . . . . . 7 class ℕ
6	5, 5	cxp 5632	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3902	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2114	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1541	. . . . . 6 class 𝑓
11		c0 4287	. . . . . . 7 class ∅
12	11	csn 4582	. . . . . 6 class {∅}
13	10, 12	cdif 3900	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6496	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5634	. . . . 5 class dom 𝑓
16		cfz 13437	. . . . . 6 class ...
17	3, 16	cfv 6502	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3903	. . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
19	8, 14, 18	w3a 1087	. . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
20	19, 9, 2	copab 5162	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1542	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17089  isstruct2  17090