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Definition df-struct 11975
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 5141, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 11986: 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 3726). (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 11969 . 2 class Struct
2 vx . . . . . 6 setvar 𝑥
32cv 1330 . . . . 5 class 𝑥
4 cle 7813 . . . . . 6 class
5 cn 8732 . . . . . . 7 class
65, 5cxp 4537 . . . . . 6 class (ℕ × ℕ)
74, 6cin 3070 . . . . 5 class ( ≤ ∩ (ℕ × ℕ))
83, 7wcel 1480 . . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9 vf . . . . . . 7 setvar 𝑓
109cv 1330 . . . . . 6 class 𝑓
11 c0 3363 . . . . . . 7 class
1211csn 3527 . . . . . 6 class {∅}
1310, 12cdif 3068 . . . . 5 class (𝑓 ∖ {∅})
1413wfun 5117 . . . 4 wff Fun (𝑓 ∖ {∅})
1510cdm 4539 . . . . 5 class dom 𝑓
16 cfz 9802 . . . . . 6 class ...
173, 16cfv 5123 . . . . 5 class (...‘𝑥)
1815, 17wss 3071 . . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
198, 14, 18w3a 962 . . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
2019, 9, 2copab 3988 . 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
211, 20wceq 1331 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  11982  isstruct2im  11983  isstruct2r  11984
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