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Definition df-struct 16475
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 6367, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16485: 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 4820). This is used critically in strle1 16582, strle2 16583, strle3 16584 and strleun 16581 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 16633 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 16634, 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 20487. (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 16469 . 2 class Struct
2 vx . . . . . 6 setvar 𝑥
32cv 1527 . . . . 5 class 𝑥
4 cle 10665 . . . . . 6 class
5 cn 11627 . . . . . . 7 class
65, 5cxp 5547 . . . . . 6 class (ℕ × ℕ)
74, 6cin 3934 . . . . 5 class ( ≤ ∩ (ℕ × ℕ))
83, 7wcel 2105 . . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9 vf . . . . . . 7 setvar 𝑓
109cv 1527 . . . . . 6 class 𝑓
11 c0 4290 . . . . . . 7 class
1211csn 4559 . . . . . 6 class {∅}
1310, 12cdif 3932 . . . . 5 class (𝑓 ∖ {∅})
1413wfun 6343 . . . 4 wff Fun (𝑓 ∖ {∅})
1510cdm 5549 . . . . 5 class dom 𝑓
16 cfz 12882 . . . . . 6 class ...
173, 16cfv 6349 . . . . 5 class (...‘𝑥)
1815, 17wss 3935 . . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
198, 14, 18w3a 1079 . . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
2019, 9, 2copab 5120 . 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
211, 20wceq 1528 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  16482  isstruct2  16483
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