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Definition df-struct 15643
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. (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 15637 . 2 class Struct
2 vx . . . . . 6 setvar 𝑥
32cv 1473 . . . . 5 class 𝑥
4 cle 9931 . . . . . 6 class
5 cn 10867 . . . . . . 7 class
65, 5cxp 5026 . . . . . 6 class (ℕ × ℕ)
74, 6cin 3538 . . . . 5 class ( ≤ ∩ (ℕ × ℕ))
83, 7wcel 1976 . . . 4 wff 𝑥 ∈ ( ≤ ∩ (ℕ × ℕ))
9 vf . . . . . . 7 setvar 𝑓
109cv 1473 . . . . . 6 class 𝑓
11 c0 3873 . . . . . . 7 class
1211csn 4124 . . . . . 6 class {∅}
1310, 12cdif 3536 . . . . 5 class (𝑓 ∖ {∅})
1413wfun 5784 . . . 4 wff Fun (𝑓 ∖ {∅})
1510cdm 5028 . . . . 5 class dom 𝑓
16 cfz 12152 . . . . . 6 class ...
173, 16cfv 5790 . . . . 5 class (...‘𝑥)
1815, 17wss 3539 . . . 4 wff dom 𝑓 ⊆ (...‘𝑥)
198, 14, 18w3a 1030 . . 3 wff (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))
2019, 9, 2copab 4636 . 2 class {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
211, 20wceq 1474 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  15649  isstruct2  15650
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