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Definition df-struct 17194
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 6596, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17198: 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 4920). This is used critically in strle1 17205, strle2 17206, strle3 17207 and strleun 17204 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17395 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 17396, 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 21401. (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 17193 . 2 class Struct
2 vx . . . . . 6 setvar 𝑥
32cv 1536 . . . . 5 class 𝑥
4 cle 11325 . . . . . 6 class
5 cn 12293 . . . . . . 7 class
65, 5cxp 5698 . . . . . 6 class (ℕ × ℕ)
74, 6cin 3975 . . . . 5 class ( ≤ ∩ (ℕ × ℕ))
83, 7wcel 2108 . . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9 vf . . . . . . 7 setvar 𝑓
109cv 1536 . . . . . 6 class 𝑓
11 c0 4352 . . . . . . 7 class
1211csn 4648 . . . . . 6 class {∅}
1310, 12cdif 3973 . . . . 5 class (𝑓 ∖ {∅})
1413wfun 6567 . . . 4 wff Fun (𝑓 ∖ {∅})
1510cdm 5700 . . . . 5 class dom 𝑓
16 cfz 13567 . . . . . 6 class ...
173, 16cfv 6573 . . . . 5 class (...‘𝑥)
1815, 17wss 3976 . . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
198, 14, 18w3a 1087 . . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
2019, 9, 2copab 5228 . 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
211, 20wceq 1537 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  17195  isstruct2  17196
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