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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 12016 | . 2 ⊢ Fun ◡◡𝐹 |
3 | isstructim 12012 | . . . . 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 993 | . . 3 ⊢ dom 𝐹 ⊆ (𝑀...𝑁) |
6 | 4 | simp1i 991 | . . . . . 6 ⊢ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) |
7 | 6 | simp1i 991 | . . . . 5 ⊢ 𝑀 ∈ ℕ |
8 | elnnuz 9386 | . . . . 5 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ≥‘1)) | |
9 | 7, 8 | mpbi 144 | . . . 4 ⊢ 𝑀 ∈ (ℤ≥‘1) |
10 | fzss1 9874 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘1) → (𝑀...𝑁) ⊆ (1...𝑁)) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ (𝑀...𝑁) ⊆ (1...𝑁) |
12 | 5, 11 | sstri 3111 | . 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 963 ∈ wcel 1481 ∖ cdif 3073 ⊆ wss 3076 ∅c0 3368 {csn 3532 〈cop 3535 class class class wbr 3937 ◡ccnv 4546 dom cdm 4547 Fun wfun 5125 ‘cfv 5131 (class class class)co 5782 1c1 7645 ≤ cle 7825 ℕcn 8744 ℤ≥cuz 9350 ...cfz 9821 Struct cstr 11994 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-z 9079 df-uz 9351 df-fz 9822 df-struct 12000 |
This theorem is referenced by: (None) |
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