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Mirrors > Home > MPE Home > Th. List > wrdl1s1 | Structured version Visualization version GIF version |
Description: A word of length 1 is a singleton word consisting of the first symbol of the word. (Contributed by AV, 22-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.) |
Ref | Expression |
---|---|
wrdl1s1 | ⊢ (𝑆 ∈ 𝑉 → (𝑊 = 〈“𝑆”〉 ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1cl 13618 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → 〈“𝑆”〉 ∈ Word 𝑉) | |
2 | s1len 13622 | . . . . 5 ⊢ (♯‘〈“𝑆”〉) = 1 | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (♯‘〈“𝑆”〉) = 1) |
4 | s1fv 13626 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉‘0) = 𝑆) | |
5 | 1, 3, 4 | 3jca 1159 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉 ∈ Word 𝑉 ∧ (♯‘〈“𝑆”〉) = 1 ∧ (〈“𝑆”〉‘0) = 𝑆)) |
6 | eleq1 2864 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → (𝑊 ∈ Word 𝑉 ↔ 〈“𝑆”〉 ∈ Word 𝑉)) | |
7 | fveqeq2 6418 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → ((♯‘𝑊) = 1 ↔ (♯‘〈“𝑆”〉) = 1)) | |
8 | fveq1 6408 | . . . . 5 ⊢ (𝑊 = 〈“𝑆”〉 → (𝑊‘0) = (〈“𝑆”〉‘0)) | |
9 | 8 | eqeq1d 2799 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → ((𝑊‘0) = 𝑆 ↔ (〈“𝑆”〉‘0) = 𝑆)) |
10 | 6, 7, 9 | 3anbi123d 1561 | . . 3 ⊢ (𝑊 = 〈“𝑆”〉 → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆) ↔ (〈“𝑆”〉 ∈ Word 𝑉 ∧ (♯‘〈“𝑆”〉) = 1 ∧ (〈“𝑆”〉‘0) = 𝑆))) |
11 | 5, 10 | syl5ibrcom 239 | . 2 ⊢ (𝑆 ∈ 𝑉 → (𝑊 = 〈“𝑆”〉 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) |
12 | eqs1 13628 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → 𝑊 = 〈“(𝑊‘0)”〉) | |
13 | s1eq 13616 | . . . . 5 ⊢ ((𝑊‘0) = 𝑆 → 〈“(𝑊‘0)”〉 = 〈“𝑆”〉) | |
14 | 13 | eqeq2d 2807 | . . . 4 ⊢ ((𝑊‘0) = 𝑆 → (𝑊 = 〈“(𝑊‘0)”〉 ↔ 𝑊 = 〈“𝑆”〉)) |
15 | 12, 14 | syl5ibcom 237 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → ((𝑊‘0) = 𝑆 → 𝑊 = 〈“𝑆”〉)) |
16 | 15 | 3impia 1146 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆) → 𝑊 = 〈“𝑆”〉) |
17 | 11, 16 | impbid1 217 | 1 ⊢ (𝑆 ∈ 𝑉 → (𝑊 = 〈“𝑆”〉 ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ‘cfv 6099 0cc0 10222 1c1 10223 ♯chash 13366 Word cword 13530 〈“cs1 13611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-card 9049 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-n0 11577 df-z 11663 df-uz 11927 df-fz 12577 df-fzo 12717 df-hash 13367 df-word 13531 df-s1 13612 |
This theorem is referenced by: rusgrnumwwlkb0 27254 clwwlknon1 27428 |
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