Theorem isstruct2im 11567

Description: The property of being a structure with components in (1^st ‘𝑋)...(2^nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)

Assertion

Ref	Expression
isstruct2im	⊢ (𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Proof of Theorem isstruct2im

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brstruct 11566	. . . 4 ⊢ Rel Struct
2	1	brrelex12i 4495	. . 3 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∈ V ∧ 𝑋 ∈ V))
3		simpr 109	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
4	3	eleq1d 2157	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ))))
5		simpl 108	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑓 = 𝐹)
6	5	difeq1d 3120	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅}))
7	6	funeqd 5052	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})))
8	5	dmeqd 4653	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → dom 𝑓 = dom 𝐹)
9	3	fveq2d 5324	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋))
10	8, 9	sseq12d 3058	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋)))
11	4, 7, 10	3anbi123d 1249	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
12		df-struct 11559	. . . 4 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
13	11, 12	brabga 4102	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝑋 ∈ V) → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
14	2, 13	syl 14	. 2 ⊢ (𝐹 Struct 𝑋 → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
15	14	ibi 175	1 ⊢ (𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 925   = wceq 1290   ∈ wcel 1439  Vcvv 2622   ∖ cdif 2999   ∩ cin 3001   ⊆ wss 3002  ∅c0 3289  {csn 3452   class class class wbr 3853   × cxp 4452  dom cdm 4454  Fun wfun 5024  ‘cfv 5030   ≤ cle 7586  ℕcn 8485  ...cfz 9487   Struct cstr 11553

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2624  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fv 5038  df-struct 11559

This theorem is referenced by:  structn0fun  11570  isstructim  11571