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Mirrors > Home > MPE Home > Th. List > loopclwwlkn1b | Structured version Visualization version GIF version |
Description: The singleton word consisting of a vertex 𝑉 represents a closed walk of length 1 iff there is a loop at vertex 𝑉. (Contributed by AV, 11-Feb-2022.) |
Ref | Expression |
---|---|
loopclwwlkn1b | ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) ↔ 〈“𝑉”〉 ∈ (1 ClWWalksN 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlkn1 27349 | . 2 ⊢ (〈“𝑉”〉 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺))) | |
2 | s1fv 13629 | . . . . . . . 8 ⊢ (𝑉 ∈ (Vtx‘𝐺) → (〈“𝑉”〉‘0) = 𝑉) | |
3 | 2 | sneqd 4381 | . . . . . . 7 ⊢ (𝑉 ∈ (Vtx‘𝐺) → {(〈“𝑉”〉‘0)} = {𝑉}) |
4 | 3 | eleq1d 2864 | . . . . . 6 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺) ↔ {𝑉} ∈ (Edg‘𝐺))) |
5 | 4 | biimpcd 241 | . . . . 5 ⊢ ({(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺) → (𝑉 ∈ (Vtx‘𝐺) → {𝑉} ∈ (Edg‘𝐺))) |
6 | 5 | 3ad2ant3 1166 | . . . 4 ⊢ (((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)) → (𝑉 ∈ (Vtx‘𝐺) → {𝑉} ∈ (Edg‘𝐺))) |
7 | 6 | com12 32 | . . 3 ⊢ (𝑉 ∈ (Vtx‘𝐺) → (((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)) → {𝑉} ∈ (Edg‘𝐺))) |
8 | s1len 13625 | . . . . . 6 ⊢ (♯‘〈“𝑉”〉) = 1 | |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → (♯‘〈“𝑉”〉) = 1) |
10 | s1cl 13621 | . . . . . 6 ⊢ (𝑉 ∈ (Vtx‘𝐺) → 〈“𝑉”〉 ∈ Word (Vtx‘𝐺)) | |
11 | 10 | adantr 473 | . . . . 5 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → 〈“𝑉”〉 ∈ Word (Vtx‘𝐺)) |
12 | 2 | eqcomd 2806 | . . . . . . . 8 ⊢ (𝑉 ∈ (Vtx‘𝐺) → 𝑉 = (〈“𝑉”〉‘0)) |
13 | 12 | sneqd 4381 | . . . . . . 7 ⊢ (𝑉 ∈ (Vtx‘𝐺) → {𝑉} = {(〈“𝑉”〉‘0)}) |
14 | 13 | eleq1d 2864 | . . . . . 6 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) ↔ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺))) |
15 | 14 | biimpa 469 | . . . . 5 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)) |
16 | 9, 11, 15 | 3jca 1159 | . . . 4 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → ((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺))) |
17 | 16 | ex 402 | . . 3 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) → ((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)))) |
18 | 7, 17 | impbid 204 | . 2 ⊢ (𝑉 ∈ (Vtx‘𝐺) → (((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)) ↔ {𝑉} ∈ (Edg‘𝐺))) |
19 | 1, 18 | syl5rbb 276 | 1 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) ↔ 〈“𝑉”〉 ∈ (1 ClWWalksN 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 {csn 4369 ‘cfv 6102 (class class class)co 6879 0cc0 10225 1c1 10226 ♯chash 13369 Word cword 13533 〈“cs1 13614 Vtxcvtx 26230 Edgcedg 26281 ClWWalksN cclwwlkn 27325 |
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 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
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 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-oadd 7804 df-er 7983 df-map 8098 df-pm 8099 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-card 9052 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-n0 11580 df-xnn0 11652 df-z 11666 df-uz 11930 df-fz 12580 df-fzo 12720 df-hash 13370 df-word 13534 df-lsw 13582 df-s1 13615 df-clwwlk 27274 df-clwwlkn 27327 |
This theorem is referenced by: (None) |
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