Definition df-struct 17187

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 6589, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17191: 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 4902). This is used critically in strle1 17198, strle2 17199, strle3 17200 and strleun 17197 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17388 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 17389, 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 21402. (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 17186	. 2 class Struct
2		vx	. . . . . 6 setvar 𝑥
3	2	cv 1537	. . . . 5 class 𝑥
4		cle 11300	. . . . . 6 class ≤
5		cn 12270	. . . . . . 7 class ℕ
6	5, 5	cxp 5688	. . . . . 6 class (ℕ × ℕ)
7	4, 6	cin 3963	. . . . 5 class ( ≤ ∩ (ℕ × ℕ))
8	3, 7	wcel 2107	. . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9		vf	. . . . . . 7 setvar 𝑓
10	9	cv 1537	. . . . . 6 class 𝑓
11		c0 4340	. . . . . . 7 class ∅
12	11	csn 4632	. . . . . 6 class {∅}
13	10, 12	cdif 3961	. . . . 5 class (𝑓 ∖ {∅})
14	13	wfun 6560	. . . 4 wff Fun (𝑓 ∖ {∅})
15	10	cdm 5690	. . . . 5 class dom 𝑓
16		cfz 13550	. . . . . 6 class ...
17	3, 16	cfv 6566	. . . . 5 class (...‘𝑥)
18	15, 17	wss 3964	. . . 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 5211	. 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
21	1, 20	wceq 1538	1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}

This definition is referenced by:  brstruct  17188  isstruct2  17189