Theorem isstructim 12606

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

Proof of Theorem isstructim

Step	Hyp	Ref	Expression
1		isstruct2im 12602	. 2 ⊢ (𝐹 Struct ⟨𝑀, 𝑁⟩ → (⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩)))
2		brinxp2 4718	. . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁))
3		df-br 4026	. . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
4	2, 3	bitr3i 186	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ↔ ⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)))
5		biid 171	. . 3 ⊢ (Fun (𝐹 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅}))
6		df-ov 5909	. . . 4 ⊢ (𝑀...𝑁) = (...‘⟨𝑀, 𝑁⟩)
7	6	sseq2i 3202	. . 3 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) ↔ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))
8	4, 5, 7	3anbi123i 1190	. 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)) ↔ (⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩)))
9	1, 8	sylibr 134	1 ⊢ (𝐹 Struct ⟨𝑀, 𝑁⟩ → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)))

Syntax hints:   → wi 4   ∧ w3a 980   ∈ wcel 2160   ∖ cdif 3146   ∩ cin 3148   ⊆ wss 3149  ∅c0 3442  {csn 3614  ⟨cop 3617   class class class wbr 4025   × cxp 4649  dom cdm 4651  Fun wfun 5236  ‘cfv 5242  (class class class)co 5906   ≤ cle 8041  ℕcn 8968  ...cfz 10060   Struct cstr 12588

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2758  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-br 4026  df-opab 4087  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-iota 5203  df-fun 5244  df-fv 5250  df-ov 5909  df-struct 12594

This theorem is referenced by:  structfn  12611  strsetsid  12625  strleund  12695  strleun  12696  strext  12697