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Mirrors > Home > ILE Home > Th. List > isstructr | 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 |
---|---|
isstructr | ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct 〈𝑀, 𝑁〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brinxp2 4534 | . . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁)) | |
2 | df-br 3868 | . . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ 〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ))) | |
3 | 1, 2 | sylbb1 136 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) → 〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ))) |
4 | 3 | adantr 271 | . 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ))) |
5 | simpr1 952 | . 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → Fun (𝐹 ∖ {∅})) | |
6 | simpr2 953 | . 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 ∈ 𝑉) | |
7 | df-ov 5693 | . . . . . 6 ⊢ (𝑀...𝑁) = (...‘〈𝑀, 𝑁〉) | |
8 | 7 | sseq2i 3066 | . . . . 5 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) ↔ dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉)) |
9 | 8 | biimpi 119 | . . . 4 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) → dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉)) |
10 | 9 | 3ad2ant3 969 | . . 3 ⊢ ((Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁)) → dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉)) |
11 | 10 | adantl 272 | . 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉)) |
12 | isstruct2r 11654 | . 2 ⊢ (((〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅})) ∧ (𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉))) → 𝐹 Struct 〈𝑀, 𝑁〉) | |
13 | 4, 5, 6, 11, 12 | syl22anc 1182 | 1 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ (Fun (𝐹 ∖ {∅}) ∧ 𝐹 ∈ 𝑉 ∧ dom 𝐹 ⊆ (𝑀...𝑁))) → 𝐹 Struct 〈𝑀, 𝑁〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 927 ∈ wcel 1445 ∖ cdif 3010 ∩ cin 3012 ⊆ wss 3013 ∅c0 3302 {csn 3466 〈cop 3469 class class class wbr 3867 × cxp 4465 dom cdm 4467 Fun wfun 5043 ‘cfv 5049 (class class class)co 5690 ≤ cle 7620 ℕcn 8520 ...cfz 9573 Struct cstr 11639 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-ov 5693 df-struct 11645 |
This theorem is referenced by: strleund 11731 strleun 11732 strle1g 11733 |
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