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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 12493 | . 2 ⊢ Fun ◡◡𝐹 |
3 | isstructim 12489 | . . . . 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 1009 | . . 3 ⊢ dom 𝐹 ⊆ (𝑀...𝑁) |
6 | 4 | simp1i 1007 | . . . . . 6 ⊢ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) |
7 | 6 | simp1i 1007 | . . . . 5 ⊢ 𝑀 ∈ ℕ |
8 | elnnuz 9577 | . . . . 5 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ≥‘1)) | |
9 | 7, 8 | mpbi 145 | . . . 4 ⊢ 𝑀 ∈ (ℤ≥‘1) |
10 | fzss1 10076 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘1) → (𝑀...𝑁) ⊆ (1...𝑁)) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ (𝑀...𝑁) ⊆ (1...𝑁) |
12 | 5, 11 | sstri 3176 | . 2 ⊢ dom 𝐹 ⊆ (1...𝑁) |
13 | 2, 12 | pm3.2i 272 | 1 ⊢ (Fun ◡◡𝐹 ∧ dom 𝐹 ⊆ (1...𝑁)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∧ w3a 979 ∈ wcel 2158 ∖ cdif 3138 ⊆ wss 3141 ∅c0 3434 {csn 3604 〈cop 3607 class class class wbr 4015 ◡ccnv 4637 dom cdm 4638 Fun wfun 5222 ‘cfv 5228 (class class class)co 5888 1c1 7825 ≤ cle 8006 ℕcn 8932 ℤ≥cuz 9541 ...cfz 10021 Struct cstr 12471 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-z 9267 df-uz 9542 df-fz 10022 df-struct 12477 |
This theorem is referenced by: (None) |
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