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Definition df-struct 16480
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 6346, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16490: 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 4791). This is used critically in strle1 16587, strle2 16588, strle3 16589 and strleun 16586 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 16638 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 16639, 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 20106. (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 16474 . 2 class Struct
2 vx . . . . . 6 setvar 𝑥
32cv 1537 . . . . 5 class 𝑥
4 cle 10669 . . . . . 6 class
5 cn 11629 . . . . . . 7 class
65, 5cxp 5521 . . . . . 6 class (ℕ × ℕ)
74, 6cin 3883 . . . . 5 class ( ≤ ∩ (ℕ × ℕ))
83, 7wcel 2112 . . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9 vf . . . . . . 7 setvar 𝑓
109cv 1537 . . . . . 6 class 𝑓
11 c0 4246 . . . . . . 7 class
1211csn 4528 . . . . . 6 class {∅}
1310, 12cdif 3881 . . . . 5 class (𝑓 ∖ {∅})
1413wfun 6322 . . . 4 wff Fun (𝑓 ∖ {∅})
1510cdm 5523 . . . . 5 class dom 𝑓
16 cfz 12889 . . . . . 6 class ...
173, 16cfv 6328 . . . . 5 class (...‘𝑥)
1815, 17wss 3884 . . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
198, 14, 18w3a 1084 . . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
2019, 9, 2copab 5095 . 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
211, 20wceq 1538 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  16487  isstruct2  16488
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