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Theorem isstructr 11988
 Description: The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
Assertion
Ref Expression
isstructr (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)

Proof of Theorem isstructr
StepHypRef Expression
1 brinxp2 4606 . . . 4 (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁))
2 df-br 3930 . . . 4 (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
31, 2sylbb1 136 . . 3 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
43adantr 274 . 2 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
5 simpr1 987 . 2 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → Fun (𝐹 ∖ {∅}))
6 simpr2 988 . 2 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹𝑉)
7 df-ov 5777 . . . . . 6 (𝑀...𝑁) = (...‘⟨𝑀, 𝑁⟩)
87sseq2i 3124 . . . . 5 (dom 𝐹 ⊆ (𝑀...𝑁) ↔ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
98biimpi 119 . . . 4 (dom 𝐹 ⊆ (𝑀...𝑁) → dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
1093ad2ant3 1004 . . 3 ((Fun (𝐹 ∖ {∅}) ∧ 𝐹𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁)) → dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
1110adantl 275 . 2 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
12 isstruct2r 11984 . 2 (((⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹𝑉 ∧ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)
134, 5, 6, 11, 12syl22anc 1217 1 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   ∈ wcel 1480   ∖ cdif 3068   ∩ cin 3070   ⊆ wss 3071  ∅c0 3363  {csn 3527  ⟨cop 3530   class class class wbr 3929   × cxp 4537  dom cdm 4539  Fun wfun 5117  ‘cfv 5123  (class class class)co 5774   ≤ cle 7813  ℕcn 8732  ...cfz 9802   Struct cstr 11969 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-ov 5777  df-struct 11975 This theorem is referenced by:  strleund  12061  strleun  12062  strle1g  12063
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