Theorem isstructr 12718

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 ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)

Proof of Theorem isstructr

Step	Hyp	Ref	Expression
1		brinxp2 4731	. . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁))
2		df-br 4035	. . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
3	1, 2	sylbb1 137	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
4	3	adantr 276	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
5		simpr1 1005	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → Fun (𝐹 ∖ {∅}))
6		simpr2 1006	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 ∈ 𝑉)
7		df-ov 5928	. . . . . 6 ⊢ (𝑀...𝑁) = (...‘⟨𝑀, 𝑁⟩)
8	7	sseq2i 3211	. . . . 5 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) ↔ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
9	8	biimpi 120	. . . 4 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) → dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
10	9	3ad2ant3 1022	. . 3 ⊢ ((Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁)) → dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
11	10	adantl 277	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
12		isstruct2r 12714	. 2 ⊢ (((⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)
13	4, 5, 6, 11, 12	syl22anc 1250	1 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   ∈ wcel 2167   ∖ cdif 3154   ∩ cin 3156   ⊆ wss 3157  ∅c0 3451  {csn 3623  ⟨cop 3626   class class class wbr 4034   × cxp 4662  dom cdm 4664  Fun wfun 5253  ‘cfv 5259  (class class class)co 5925   ≤ cle 8079  ℕcn 9007  ...cfz 10100   Struct cstr 12699

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-ov 5928  df-struct 12705

This theorem is referenced by:  strleund  12806  strleun  12807  strext  12808  strle1g  12809