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Definition df-struct 16314
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 6243, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16324: 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 4733). This is used critically in strle1 16421, strle2 16422, strle3 16423 and strleun 16420 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 16472 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 16473, which requires that the base set be a set but not any of the other components. Usually, a concrete structure like fld does not contain the empty set, and therefore is a function, see cnfldfun 20239. (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
StepHypRef Expression
1 cstr 16308 . 2 class Struct
2 vx . . . . . 6 setvar 𝑥
32cv 1521 . . . . 5 class 𝑥
4 cle 10522 . . . . . 6 class
5 cn 11486 . . . . . . 7 class
65, 5cxp 5441 . . . . . 6 class (ℕ × ℕ)
74, 6cin 3858 . . . . 5 class ( ≤ ∩ (ℕ × ℕ))
83, 7wcel 2081 . . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9 vf . . . . . . 7 setvar 𝑓
109cv 1521 . . . . . 6 class 𝑓
11 c0 4211 . . . . . . 7 class
1211csn 4472 . . . . . 6 class {∅}
1310, 12cdif 3856 . . . . 5 class (𝑓 ∖ {∅})
1413wfun 6219 . . . 4 wff Fun (𝑓 ∖ {∅})
1510cdm 5443 . . . . 5 class dom 𝑓
16 cfz 12742 . . . . . 6 class ...
173, 16cfv 6225 . . . . 5 class (...‘𝑥)
1815, 17wss 3859 . . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
198, 14, 18w3a 1080 . . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
2019, 9, 2copab 5024 . 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
211, 20wceq 1522 1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
Colors of variables: wff setvar class
This definition is referenced by:  brstruct  16321  isstruct2  16322
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