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Mirrors > Home > MPE Home > Th. List > isstruct | Structured version Visualization version GIF version |
Description: The property of being a structure with components in 𝑀...𝑁. (Contributed by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
isstruct | ⊢ (𝐹 Struct 〈𝑀, 𝑁〉 ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isstruct2 17191 | . 2 ⊢ (𝐹 Struct 〈𝑀, 𝑁〉 ↔ (〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉))) | |
2 | df-3an 1089 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑀 ≤ 𝑁)) | |
3 | brinxp2 5776 | . . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑀 ≤ 𝑁)) | |
4 | df-br 5170 | . . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ 〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ))) | |
5 | 2, 3, 4 | 3bitr2i 299 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ↔ 〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ))) |
6 | biid 261 | . . 3 ⊢ (Fun (𝐹 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})) | |
7 | df-ov 7448 | . . . 4 ⊢ (𝑀...𝑁) = (...‘〈𝑀, 𝑁〉) | |
8 | 7 | sseq2i 4032 | . . 3 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) ↔ dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉)) |
9 | 5, 6, 8 | 3anbi123i 1155 | . 2 ⊢ (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁)) ↔ (〈𝑀, 𝑁〉 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘〈𝑀, 𝑁〉))) |
10 | 1, 9 | bitr4i 278 | 1 ⊢ (𝐹 Struct 〈𝑀, 𝑁〉 ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (𝑀...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2103 ∖ cdif 3967 ∩ cin 3969 ⊆ wss 3970 ∅c0 4347 {csn 4648 〈cop 4654 class class class wbr 5169 × cxp 5697 dom cdm 5699 Fun wfun 6566 ‘cfv 6572 (class class class)co 7445 ≤ cle 11321 ℕcn 12289 ...cfz 13563 Struct cstr 17188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-struct 17189 |
This theorem is referenced by: structfn 17198 strleun 17199 strle1 17200 setsstruct 17218 |
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