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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 14559 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → 〈“𝑆”〉 ∈ Word 𝑉) | |
2 | s1len 14563 | . . . . 5 ⊢ (♯‘〈“𝑆”〉) = 1 | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (♯‘〈“𝑆”〉) = 1) |
4 | s1fv 14567 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉‘0) = 𝑆) | |
5 | 1, 3, 4 | 3jca 1127 | . . 3 ⊢ (𝑆 ∈ 𝑉 → (〈“𝑆”〉 ∈ Word 𝑉 ∧ (♯‘〈“𝑆”〉) = 1 ∧ (〈“𝑆”〉‘0) = 𝑆)) |
6 | eleq1 2820 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → (𝑊 ∈ Word 𝑉 ↔ 〈“𝑆”〉 ∈ Word 𝑉)) | |
7 | fveqeq2 6900 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → ((♯‘𝑊) = 1 ↔ (♯‘〈“𝑆”〉) = 1)) | |
8 | fveq1 6890 | . . . . 5 ⊢ (𝑊 = 〈“𝑆”〉 → (𝑊‘0) = (〈“𝑆”〉‘0)) | |
9 | 8 | eqeq1d 2733 | . . . 4 ⊢ (𝑊 = 〈“𝑆”〉 → ((𝑊‘0) = 𝑆 ↔ (〈“𝑆”〉‘0) = 𝑆)) |
10 | 6, 7, 9 | 3anbi123d 1435 | . . 3 ⊢ (𝑊 = 〈“𝑆”〉 → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆) ↔ (〈“𝑆”〉 ∈ Word 𝑉 ∧ (♯‘〈“𝑆”〉) = 1 ∧ (〈“𝑆”〉‘0) = 𝑆))) |
11 | 5, 10 | syl5ibrcom 246 | . 2 ⊢ (𝑆 ∈ 𝑉 → (𝑊 = 〈“𝑆”〉 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) |
12 | eqs1 14569 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → 𝑊 = 〈“(𝑊‘0)”〉) | |
13 | s1eq 14557 | . . . . 5 ⊢ ((𝑊‘0) = 𝑆 → 〈“(𝑊‘0)”〉 = 〈“𝑆”〉) | |
14 | 13 | eqeq2d 2742 | . . . 4 ⊢ ((𝑊‘0) = 𝑆 → (𝑊 = 〈“(𝑊‘0)”〉 ↔ 𝑊 = 〈“𝑆”〉)) |
15 | 12, 14 | syl5ibcom 244 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → ((𝑊‘0) = 𝑆 → 𝑊 = 〈“𝑆”〉)) |
16 | 15 | 3impia 1116 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆) → 𝑊 = 〈“𝑆”〉) |
17 | 11, 16 | impbid1 224 | 1 ⊢ (𝑆 ∈ 𝑉 → (𝑊 = 〈“𝑆”〉 ↔ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1 ∧ (𝑊‘0) = 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 0cc0 11116 1c1 11117 ♯chash 14297 Word cword 14471 〈“cs1 14552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-hash 14298 df-word 14472 df-s1 14553 |
This theorem is referenced by: rusgrnumwwlkb0 29660 clwwlknon1 29785 |
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