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Mirrors > Home > ILE Home > Th. List > isstruct2im | GIF version |
Description: The property of being a structure with components in (1st ‘𝑋)...(2nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.) |
Ref | Expression |
---|---|
isstruct2im | ⊢ (𝐹 Struct 𝑋 → (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brstruct 11971 | . . . 4 ⊢ Rel Struct | |
2 | 1 | brrelex12i 4581 | . . 3 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∈ V ∧ 𝑋 ∈ V)) |
3 | simpr 109 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
4 | 3 | eleq1d 2208 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)))) |
5 | simpl 108 | . . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑓 = 𝐹) | |
6 | 5 | difeq1d 3193 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅})) |
7 | 6 | funeqd 5145 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅}))) |
8 | 5 | dmeqd 4741 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → dom 𝑓 = dom 𝐹) |
9 | 3 | fveq2d 5425 | . . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋)) |
10 | 8, 9 | sseq12d 3128 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋))) |
11 | 4, 7, 10 | 3anbi123d 1290 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))) |
12 | df-struct 11964 | . . . 4 ⊢ Struct = {〈𝑓, 𝑥〉 ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))} | |
13 | 11, 12 | brabga 4186 | . . 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))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∖ cdif 3068 ∩ cin 3070 ⊆ wss 3071 ∅c0 3363 {csn 3527 class class class wbr 3929 × cxp 4537 dom cdm 4539 Fun wfun 5117 ‘cfv 5123 ≤ cle 7804 ℕcn 8723 ...cfz 9793 Struct cstr 11958 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-struct 11964 |
This theorem is referenced by: structn0fun 11975 isstructim 11976 |
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