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| Mirrors > Home > MPE Home > Th. List > strle1 | Structured version Visualization version GIF version | ||
| Description: Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| strle1.i | ⊢ 𝐼 ∈ ℕ | 
| strle1.a | ⊢ 𝐴 = 𝐼 | 
| Ref | Expression | 
|---|---|
| strle1 | ⊢ {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | strle1.i | . . 3 ⊢ 𝐼 ∈ ℕ | |
| 2 | 1 | nnrei 12276 | . . . 4 ⊢ 𝐼 ∈ ℝ | 
| 3 | 2 | leidi 11798 | . . 3 ⊢ 𝐼 ≤ 𝐼 | 
| 4 | 1, 1, 3 | 3pm3.2i 1339 | . 2 ⊢ (𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) | 
| 5 | difss 4135 | . . . 4 ⊢ ({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} | |
| 6 | strle1.a | . . . . . 6 ⊢ 𝐴 = 𝐼 | |
| 7 | 6, 1 | eqeltri 2836 | . . . . 5 ⊢ 𝐴 ∈ ℕ | 
| 8 | funsng 6616 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑋 ∈ V) → Fun {〈𝐴, 𝑋〉}) | |
| 9 | 7, 8 | mpan 690 | . . . 4 ⊢ (𝑋 ∈ V → Fun {〈𝐴, 𝑋〉}) | 
| 10 | funss 6584 | . . . 4 ⊢ (({〈𝐴, 𝑋〉} ∖ {∅}) ⊆ {〈𝐴, 𝑋〉} → (Fun {〈𝐴, 𝑋〉} → Fun ({〈𝐴, 𝑋〉} ∖ {∅}))) | |
| 11 | 5, 9, 10 | mpsyl 68 | . . 3 ⊢ (𝑋 ∈ V → Fun ({〈𝐴, 𝑋〉} ∖ {∅})) | 
| 12 | fun0 6630 | . . . 4 ⊢ Fun ∅ | |
| 13 | opprc2 4897 | . . . . . . . 8 ⊢ (¬ 𝑋 ∈ V → 〈𝐴, 𝑋〉 = ∅) | |
| 14 | 13 | sneqd 4637 | . . . . . . 7 ⊢ (¬ 𝑋 ∈ V → {〈𝐴, 𝑋〉} = {∅}) | 
| 15 | 14 | difeq1d 4124 | . . . . . 6 ⊢ (¬ 𝑋 ∈ V → ({〈𝐴, 𝑋〉} ∖ {∅}) = ({∅} ∖ {∅})) | 
| 16 | difid 4375 | . . . . . 6 ⊢ ({∅} ∖ {∅}) = ∅ | |
| 17 | 15, 16 | eqtrdi 2792 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → ({〈𝐴, 𝑋〉} ∖ {∅}) = ∅) | 
| 18 | 17 | funeqd 6587 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (Fun ({〈𝐴, 𝑋〉} ∖ {∅}) ↔ Fun ∅)) | 
| 19 | 12, 18 | mpbiri 258 | . . 3 ⊢ (¬ 𝑋 ∈ V → Fun ({〈𝐴, 𝑋〉} ∖ {∅})) | 
| 20 | 11, 19 | pm2.61i 182 | . 2 ⊢ Fun ({〈𝐴, 𝑋〉} ∖ {∅}) | 
| 21 | dmsnopss 6233 | . . 3 ⊢ dom {〈𝐴, 𝑋〉} ⊆ {𝐴} | |
| 22 | 6 | sneqi 4636 | . . . 4 ⊢ {𝐴} = {𝐼} | 
| 23 | 1 | nnzi 12643 | . . . . 5 ⊢ 𝐼 ∈ ℤ | 
| 24 | fzsn 13607 | . . . . 5 ⊢ (𝐼 ∈ ℤ → (𝐼...𝐼) = {𝐼}) | |
| 25 | 23, 24 | ax-mp 5 | . . . 4 ⊢ (𝐼...𝐼) = {𝐼} | 
| 26 | 22, 25 | eqtr4i 2767 | . . 3 ⊢ {𝐴} = (𝐼...𝐼) | 
| 27 | 21, 26 | sseqtri 4031 | . 2 ⊢ dom {〈𝐴, 𝑋〉} ⊆ (𝐼...𝐼) | 
| 28 | isstruct 17190 | . 2 ⊢ ({〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉 ↔ ((𝐼 ∈ ℕ ∧ 𝐼 ∈ ℕ ∧ 𝐼 ≤ 𝐼) ∧ Fun ({〈𝐴, 𝑋〉} ∖ {∅}) ∧ dom {〈𝐴, 𝑋〉} ⊆ (𝐼...𝐼))) | |
| 29 | 4, 20, 27, 28 | mpbir3an 1341 | 1 ⊢ {〈𝐴, 𝑋〉} Struct 〈𝐼, 𝐼〉 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∖ cdif 3947 ⊆ wss 3950 ∅c0 4332 {csn 4625 〈cop 4631 class class class wbr 5142 dom cdm 5684 Fun wfun 6554 (class class class)co 7432 ≤ cle 11297 ℕcn 12267 ℤcz 12615 ...cfz 13548 Struct cstr 17184 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-struct 17185 | 
| This theorem is referenced by: strle2 17197 strle3 17198 1strstr 17262 1strstr1 17263 srngstr 17354 lmodstr 17370 phlstr 17391 cnfldstr 21367 cnfldstrOLD 21382 | 
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