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Definition df-struct 16668
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 6376, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 16678: 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 4793). This is used critically in strle1 16776, strle2 16777, strle3 16778 and strleun 16775 to avoid sethood hypotheses on the "payload" sets: without this, ipsstr 16827 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 16828, 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 20329. (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 16662 . 2 class Struct
2 vx . . . . . 6 setvar 𝑥
32cv 1542 . . . . 5 class 𝑥
4 cle 10833 . . . . . 6 class
5 cn 11795 . . . . . . 7 class
65, 5cxp 5534 . . . . . 6 class (ℕ × ℕ)
74, 6cin 3852 . . . . 5 class ( ≤ ∩ (ℕ × ℕ))
83, 7wcel 2112 . . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9 vf . . . . . . 7 setvar 𝑓
109cv 1542 . . . . . 6 class 𝑓
11 c0 4223 . . . . . . 7 class
1211csn 4527 . . . . . 6 class {∅}
1310, 12cdif 3850 . . . . 5 class (𝑓 ∖ {∅})
1413wfun 6352 . . . 4 wff Fun (𝑓 ∖ {∅})
1510cdm 5536 . . . . 5 class dom 𝑓
16 cfz 13060 . . . . . 6 class ...
173, 16cfv 6358 . . . . 5 class (...‘𝑥)
1815, 17wss 3853 . . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
198, 14, 18w3a 1089 . . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
2019, 9, 2copab 5101 . 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
211, 20wceq 1543 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  16675  isstruct2  16676
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