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Theorem isstruct2im 12628
Description: The property of being a structure with components in (1st𝑋)...(2nd𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
Assertion
Ref Expression
isstruct2im (𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Proof of Theorem isstruct2im
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brstruct 12627 . . . 4 Rel Struct
21brrelex12i 4701 . . 3 (𝐹 Struct 𝑋 → (𝐹 ∈ V ∧ 𝑋 ∈ V))
3 simpr 110 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑥 = 𝑋)
43eleq1d 2262 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ))))
5 simpl 109 . . . . . . 7 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑓 = 𝐹)
65difeq1d 3276 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅}))
76funeqd 5276 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})))
85dmeqd 4864 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → dom 𝑓 = dom 𝐹)
93fveq2d 5558 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋))
108, 9sseq12d 3210 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋)))
114, 7, 103anbi123d 1323 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
12 df-struct 12620 . . . 4 Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
1311, 12brabga 4294 . . 3 ((𝐹 ∈ V ∧ 𝑋 ∈ V) → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
142, 13syl 14 . 2 (𝐹 Struct 𝑋 → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
1514ibi 176 1 (𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980   = wceq 1364  wcel 2164  Vcvv 2760  cdif 3150  cin 3152  wss 3153  c0 3446  {csn 3618   class class class wbr 4029   × cxp 4657  dom cdm 4659  Fun wfun 5248  cfv 5254  cle 8055  cn 8982  ...cfz 10074   Struct cstr 12614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-struct 12620
This theorem is referenced by:  structn0fun  12631  isstructim  12632
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