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Definition df-struct 12517
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 5253, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 12528: 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 3814). (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 12511 . 2 class Struct
2 vx . . . . . 6 setvar 𝑥
32cv 1363 . . . . 5 class 𝑥
4 cle 8024 . . . . . 6 class
5 cn 8950 . . . . . . 7 class
65, 5cxp 4642 . . . . . 6 class (ℕ × ℕ)
74, 6cin 3143 . . . . 5 class ( ≤ ∩ (ℕ × ℕ))
83, 7wcel 2160 . . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9 vf . . . . . . 7 setvar 𝑓
109cv 1363 . . . . . 6 class 𝑓
11 c0 3437 . . . . . . 7 class
1211csn 3607 . . . . . 6 class {∅}
1310, 12cdif 3141 . . . . 5 class (𝑓 ∖ {∅})
1413wfun 5229 . . . 4 wff Fun (𝑓 ∖ {∅})
1510cdm 4644 . . . . 5 class dom 𝑓
16 cfz 10040 . . . . . 6 class ...
173, 16cfv 5235 . . . . 5 class (...‘𝑥)
1815, 17wss 3144 . . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
198, 14, 18w3a 980 . . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
2019, 9, 2copab 4078 . 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
211, 20wceq 1364 1 wff Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
Colors of variables: wff set class
This definition is referenced by:  brstruct  12524  isstruct2im  12525  isstruct2r  12526
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