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Mirrors > Home > ILE Home > Th. List > structfn | GIF version |
Description: Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
structfn.1 | ⊢ 𝐹 Struct 〈𝑀, 𝑁〉 |
Ref | Expression |
---|---|
structfn | ⊢ (Fun ◡◡𝐹 ∧ dom 𝐹 ⊆ (1...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | structfn.1 | . . 3 ⊢ 𝐹 Struct 〈𝑀, 𝑁〉 | |
2 | 1 | structfun 11904 | . 2 ⊢ Fun ◡◡𝐹 |
3 | isstructim 11900 | . . . . 5 ⊢ (𝐹 Struct 〈𝑀, 𝑁〉 → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁))) | |
4 | 1, 3 | ax-mp 5 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)) |
5 | 4 | simp3i 977 | . . 3 ⊢ dom 𝐹 ⊆ (𝑀...𝑁) |
6 | 4 | simp1i 975 | . . . . . 6 ⊢ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) |
7 | 6 | simp1i 975 | . . . . 5 ⊢ 𝑀 ∈ ℕ |
8 | elnnuz 9330 | . . . . 5 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ≥‘1)) | |
9 | 7, 8 | mpbi 144 | . . . 4 ⊢ 𝑀 ∈ (ℤ≥‘1) |
10 | fzss1 9811 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘1) → (𝑀...𝑁) ⊆ (1...𝑁)) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ (𝑀...𝑁) ⊆ (1...𝑁) |
12 | 5, 11 | sstri 3076 | . 2 ⊢ dom 𝐹 ⊆ (1...𝑁) |
13 | 2, 12 | pm3.2i 270 | 1 ⊢ (Fun ◡◡𝐹 ∧ dom 𝐹 ⊆ (1...𝑁)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∧ w3a 947 ∈ wcel 1465 ∖ cdif 3038 ⊆ wss 3041 ∅c0 3333 {csn 3497 〈cop 3500 class class class wbr 3899 ◡ccnv 4508 dom cdm 4509 Fun wfun 5087 ‘cfv 5093 (class class class)co 5742 1c1 7589 ≤ cle 7769 ℕcn 8688 ℤ≥cuz 9294 ...cfz 9758 Struct cstr 11882 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-z 9023 df-uz 9295 df-fz 9759 df-struct 11888 |
This theorem is referenced by: (None) |
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