Theorem isstructr 12409

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 4671	. . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁))
2		df-br 3983	. . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
3	1, 2	sylbb1 136	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
4	3	adantr 274	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
5		simpr1 993	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → Fun (𝐹 ∖ {∅}))
6		simpr2 994	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 ∈ 𝑉)
7		df-ov 5845	. . . . . 6 ⊢ (𝑀...𝑁) = (...‘⟨𝑀, 𝑁⟩)
8	7	sseq2i 3169	. . . . 5 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) ↔ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
9	8	biimpi 119	. . . 4 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) → dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
10	9	3ad2ant3 1010	. . 3 ⊢ ((Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁)) → dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
11	10	adantl 275	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
12		isstruct2r 12405	. 2 ⊢ (((⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)
13	4, 5, 6, 11, 12	syl22anc 1229	1 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct ⟨𝑀, 𝑁⟩)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 968   ∈ wcel 2136   ∖ cdif 3113   ∩ cin 3115   ⊆ wss 3116  ∅c0 3409  {csn 3576  ⟨cop 3579   class class class wbr 3982   × cxp 4602  dom cdm 4604  Fun wfun 5182  ‘cfv 5188  (class class class)co 5842   ≤ cle 7934  ℕcn 8857  ...cfz 9944   Struct cstr 12390

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-struct 12396

This theorem is referenced by:  strleund  12483  strleun  12484  strle1g  12485