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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 12610 | . 2 ⊢ Fun ◡◡𝐹 |
3 | isstructim 12606 | . . . . 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 1010 | . . 3 ⊢ dom 𝐹 ⊆ (𝑀...𝑁) |
6 | 4 | simp1i 1008 | . . . . . 6 ⊢ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) |
7 | 6 | simp1i 1008 | . . . . 5 ⊢ 𝑀 ∈ ℕ |
8 | elnnuz 9615 | . . . . 5 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ≥‘1)) | |
9 | 7, 8 | mpbi 145 | . . . 4 ⊢ 𝑀 ∈ (ℤ≥‘1) |
10 | fzss1 10115 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘1) → (𝑀...𝑁) ⊆ (1...𝑁)) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ (𝑀...𝑁) ⊆ (1...𝑁) |
12 | 5, 11 | sstri 3184 | . 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 980 ∈ wcel 2160 ∖ cdif 3146 ⊆ wss 3149 ∅c0 3442 {csn 3614 〈cop 3617 class class class wbr 4025 ◡ccnv 4650 dom cdm 4651 Fun wfun 5236 ‘cfv 5242 (class class class)co 5906 1c1 7859 ≤ cle 8041 ℕcn 8968 ℤ≥cuz 9578 ...cfz 10060 Struct cstr 12588 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-addcom 7958 ax-addass 7960 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-0id 7966 ax-rnegex 7967 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-ltadd 7974 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2758 df-sbc 2982 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-br 4026 df-opab 4087 df-mpt 4088 df-id 4318 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-inn 8969 df-z 9304 df-uz 9579 df-fz 10061 df-struct 12594 |
This theorem is referenced by: (None) |
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