Theorem isstructr 13090

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 4791	. . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁))
2		df-br 4087	. . . 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 1027	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → Fun (𝐹 ∖ {∅}))
6		simpr2 1028	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 ∈ 𝑉)
7		df-ov 6016	. . . . . 6 ⊢ (𝑀...𝑁) = (...‘⟨𝑀, 𝑁⟩)
8	7	sseq2i 3252	. . . . 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 1044	. . 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 13086	. 2 ⊢ (((⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)
13	4, 5, 6, 11, 12	syl22anc 1272	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 1002   ∈ wcel 2200   ∖ cdif 3195   ∩ cin 3197   ⊆ wss 3198  ∅c0 3492  {csn 3667  ⟨cop 3670   class class class wbr 4086   × cxp 4721  dom cdm 4723  Fun wfun 5318  ‘cfv 5324  (class class class)co 6013   ≤ cle 8208  ℕcn 9136  ...cfz 10236   Struct cstr 13071

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-struct 13077

This theorem is referenced by:  strleund  13179  strleun  13180  strext  13181  strle1g  13182