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Mirrors > Home > ILE Home > Th. List > isstructim | GIF version |
Description: The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.) |
Ref | Expression |
---|---|
isstructim | ⊢ (𝐹 Struct 〈𝑀, 𝑁〉 → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isstruct2im 12628 | . 2 ⊢ (𝐹 Struct 〈𝑀, 𝑁〉 → (〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉))) | |
2 | brinxp2 4726 | . . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁)) | |
3 | df-br 4030 | . . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ 〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ))) | |
4 | 2, 3 | bitr3i 186 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ↔ 〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ))) |
5 | biid 171 | . . 3 ⊢ (Fun (𝐹 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})) | |
6 | df-ov 5921 | . . . 4 ⊢ (𝑀...𝑁) = (...‘〈𝑀, 𝑁〉) | |
7 | 6 | sseq2i 3206 | . . 3 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) ↔ dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉)) |
8 | 4, 5, 7 | 3anbi123i 1190 | . 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)) ↔ (〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉))) |
9 | 1, 8 | sylibr 134 | 1 ⊢ (𝐹 Struct 〈𝑀, 𝑁〉 → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 980 ∈ wcel 2164 ∖ cdif 3150 ∩ cin 3152 ⊆ wss 3153 ∅c0 3446 {csn 3618 〈cop 3621 class class class wbr 4029 × cxp 4657 dom cdm 4659 Fun wfun 5248 ‘cfv 5254 (class class class)co 5918 ≤ cle 8055 ℕcn 8982 ...cfz 10074 Struct cstr 12614 |
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-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-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-ov 5921 df-struct 12620 |
This theorem is referenced by: structfn 12637 strsetsid 12651 strleund 12721 strleun 12722 strext 12723 |
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