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Definition df-struct 17149
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 6577, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 17153: 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 17160, strle2 17161, strle3 17162 and strleun 17159 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 17350 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 17351, 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 21357. (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 17148 . 2 class Struct
2 vx . . . . . 6 setvar 𝑥
32cv 1533 . . . . 5 class 𝑥
4 cle 11299 . . . . . 6 class
5 cn 12264 . . . . . . 7 class
65, 5cxp 5680 . . . . . 6 class (ℕ × ℕ)
74, 6cin 3946 . . . . 5 class ( ≤ ∩ (ℕ × ℕ))
83, 7wcel 2099 . . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9 vf . . . . . . 7 setvar 𝑓
109cv 1533 . . . . . 6 class 𝑓
11 c0 4325 . . . . . . 7 class
1211csn 4633 . . . . . 6 class {∅}
1310, 12cdif 3944 . . . . 5 class (𝑓 ∖ {∅})
1413wfun 6548 . . . 4 wff Fun (𝑓 ∖ {∅})
1510cdm 5682 . . . . 5 class dom 𝑓
16 cfz 13538 . . . . . 6 class ...
173, 16cfv 6554 . . . . 5 class (...‘𝑥)
1815, 17wss 3947 . . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
198, 14, 18w3a 1084 . . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
2019, 9, 2copab 5215 . 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
211, 20wceq 1534 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  17150  isstruct2  17151
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