Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 11958 | . 2 ⊢ (𝐹 Struct 〈𝑀, 𝑁〉 → (〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉))) | |
2 | brinxp2 4601 | . . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁)) | |
3 | df-br 3925 | . . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ 〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ))) | |
4 | 2, 3 | bitr3i 185 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ↔ 〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ))) |
5 | biid 170 | . . 3 ⊢ (Fun (𝐹 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})) | |
6 | df-ov 5770 | . . . 4 ⊢ (𝑀...𝑁) = (...‘〈𝑀, 𝑁〉) | |
7 | 6 | sseq2i 3119 | . . 3 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) ↔ dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉)) |
8 | 4, 5, 7 | 3anbi123i 1170 | . 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 133 | 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 962 ∈ wcel 1480 ∖ cdif 3063 ∩ cin 3065 ⊆ wss 3066 ∅c0 3358 {csn 3522 〈cop 3525 class class class wbr 3924 × cxp 4532 dom cdm 4534 Fun wfun 5112 ‘cfv 5118 (class class class)co 5767 ≤ cle 7794 ℕcn 8713 ...cfz 9783 Struct cstr 11944 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-ov 5770 df-struct 11950 |
This theorem is referenced by: structfn 11967 strsetsid 11981 strleund 12036 strleun 12037 |
Copyright terms: Public domain | W3C validator |