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Theorem isstruct2im 11978
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 11977 . . . 4 Rel Struct
21brrelex12i 4581 . . 3 (𝐹 Struct 𝑋 → (𝐹 ∈ V ∧ 𝑋 ∈ V))
3 simpr 109 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑥 = 𝑋)
43eleq1d 2208 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ))))
5 simpl 108 . . . . . . 7 ((𝑓 = 𝐹𝑥 = 𝑋) → 𝑓 = 𝐹)
65difeq1d 3193 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅}))
76funeqd 5145 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})))
85dmeqd 4741 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → dom 𝑓 = dom 𝐹)
93fveq2d 5425 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋))
108, 9sseq12d 3128 . . . . 5 ((𝑓 = 𝐹𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋)))
114, 7, 103anbi123d 1290 . . . 4 ((𝑓 = 𝐹𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
12 df-struct 11970 . . . 4 Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
1311, 12brabga 4186 . . 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 175 1 (𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  Vcvv 2686  cdif 3068  cin 3070  wss 3071  c0 3363  {csn 3527   class class class wbr 3929   × cxp 4537  dom cdm 4539  Fun wfun 5117  cfv 5123  cle 7808  cn 8727  ...cfz 9797   Struct cstr 11964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-struct 11970
This theorem is referenced by:  structn0fun  11981  isstructim  11982
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