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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 17091 | . 2 ⊢ (𝐹 Struct ⟨𝑀, 𝑁⟩ ↔ (⟨𝑀, 𝑁⟩ ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩))) | |
2 | df-3an 1086 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁) ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑀 ≤ 𝑁)) | |
3 | brinxp2 5746 | . . . 4 ⊢ (𝑀( ≤ ∩ (ℕ × ℕ))𝑁 ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑀 ≤ 𝑁)) | |
4 | df-br 5142 | . . . 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 7408 | . . . 4 ⊢ (𝑀...𝑁) = (...‘⟨𝑀, 𝑁⟩) | |
8 | 7 | sseq2i 4006 | . . 3 ⊢ (dom 𝐹 ⊆ (𝑀...𝑁) ↔ dom 𝐹 ⊆ (...‘⟨𝑀, 𝑁⟩)) |
9 | 5, 6, 8 | 3anbi123i 1152 | . 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 205 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 ∖ cdif 3940 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 {csn 4623 ⟨cop 4629 class class class wbr 5141 × cxp 5667 dom cdm 5669 Fun wfun 6531 ‘cfv 6537 (class class class)co 7405 ≤ cle 11253 ℕcn 12216 ...cfz 13490 Struct cstr 17088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 |
This theorem is referenced by: structfn 17098 strleun 17099 strle1 17100 setsstruct 17118 |
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