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Mirrors > Home > MPE Home > Th. List > wwlksn0 | Structured version Visualization version GIF version |
Description: A walk of length 0 is represented by a singleton word. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.) (Proof shortened by AV, 21-May-2021.) |
Ref | Expression |
---|---|
wwlkssswrd.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
wwlksn0 | ⊢ (𝑊 ∈ (0 WWalksN 𝐺) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdl1exs1 14314 | . 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ∃𝑣 ∈ (Vtx‘𝐺)𝑊 = 〈“𝑣”〉) | |
2 | fveqeq2 6778 | . . 3 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) = 1 ↔ (♯‘𝑊) = 1)) | |
3 | wwlksn0s 28220 | . . 3 ⊢ (0 WWalksN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1} | |
4 | 2, 3 | elrab2 3629 | . 2 ⊢ (𝑊 ∈ (0 WWalksN 𝐺) ↔ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1)) |
5 | wwlkssswrd.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
6 | 5 | rexeqi 3346 | . 2 ⊢ (∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉 ↔ ∃𝑣 ∈ (Vtx‘𝐺)𝑊 = 〈“𝑣”〉) |
7 | 1, 4, 6 | 3imtr4i 292 | 1 ⊢ (𝑊 ∈ (0 WWalksN 𝐺) → ∃𝑣 ∈ 𝑉 𝑊 = 〈“𝑣”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 ‘cfv 6431 (class class class)co 7269 0cc0 10870 1c1 10871 ♯chash 14040 Word cword 14213 〈“cs1 14296 Vtxcvtx 27362 WWalksN cwwlksn 28185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-n0 12232 df-xnn0 12304 df-z 12318 df-uz 12580 df-fz 13237 df-fzo 13380 df-hash 14041 df-word 14214 df-s1 14297 df-wwlks 28189 df-wwlksn 28190 |
This theorem is referenced by: (None) |
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