Theorem isstruct2im 12472

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 12471	. . . 4 ⊢ Rel Struct
2	1	brrelex12i 4669	. . 3 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∈ V ∧ 𝑋 ∈ V))
3		simpr 110	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
4	3	eleq1d 2246	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ))))
5		simpl 109	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑓 = 𝐹)
6	5	difeq1d 3253	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅}))
7	6	funeqd 5239	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})))
8	5	dmeqd 4830	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → dom 𝑓 = dom 𝐹)
9	3	fveq2d 5520	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋))
10	8, 9	sseq12d 3187	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋)))
11	4, 7, 10	3anbi123d 1312	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
12		df-struct 12464	. . . 4 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
13	11, 12	brabga 4265	. . 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 176	1 ⊢ (𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  Vcvv 2738   ∖ cdif 3127   ∩ cin 3129   ⊆ wss 3130  ∅c0 3423  {csn 3593   class class class wbr 4004   × cxp 4625  dom cdm 4627  Fun wfun 5211  ‘cfv 5217   ≤ cle 7993  ℕcn 8919  ...cfz 10008   Struct cstr 12458

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-struct 12464

This theorem is referenced by:  structn0fun  12475  isstructim  12476