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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 29923 | . 2 ⊢ (〈“𝑉”〉 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺))) | |
2 | s1fv 14596 | . . . . . . . 8 ⊢ (𝑉 ∈ (Vtx‘𝐺) → (〈“𝑉”〉‘0) = 𝑉) | |
3 | 2 | sneqd 4642 | . . . . . . 7 ⊢ (𝑉 ∈ (Vtx‘𝐺) → {(〈“𝑉”〉‘0)} = {𝑉}) |
4 | 3 | eleq1d 2810 | . . . . . 6 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺) ↔ {𝑉} ∈ (Edg‘𝐺))) |
5 | 4 | biimpcd 248 | . . . . 5 ⊢ ({(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺) → (𝑉 ∈ (Vtx‘𝐺) → {𝑉} ∈ (Edg‘𝐺))) |
6 | 5 | 3ad2ant3 1132 | . . . 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 14592 | . . . . . 6 ⊢ (♯‘〈“𝑉”〉) = 1 | |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → (♯‘〈“𝑉”〉) = 1) |
10 | s1cl 14588 | . . . . . 6 ⊢ (𝑉 ∈ (Vtx‘𝐺) → 〈“𝑉”〉 ∈ Word (Vtx‘𝐺)) | |
11 | 10 | adantr 479 | . . . . 5 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → 〈“𝑉”〉 ∈ Word (Vtx‘𝐺)) |
12 | 2 | eqcomd 2731 | . . . . . . . 8 ⊢ (𝑉 ∈ (Vtx‘𝐺) → 𝑉 = (〈“𝑉”〉‘0)) |
13 | 12 | sneqd 4642 | . . . . . . 7 ⊢ (𝑉 ∈ (Vtx‘𝐺) → {𝑉} = {(〈“𝑉”〉‘0)}) |
14 | 13 | eleq1d 2810 | . . . . . 6 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) ↔ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺))) |
15 | 14 | biimpa 475 | . . . . 5 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)) |
16 | 9, 11, 15 | 3jca 1125 | . . . 4 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → ((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺))) |
17 | 16 | ex 411 | . . 3 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) → ((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)))) |
18 | 7, 17 | impbid 211 | . 2 ⊢ (𝑉 ∈ (Vtx‘𝐺) → (((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)) ↔ {𝑉} ∈ (Edg‘𝐺))) |
19 | 1, 18 | bitr2id 283 | 1 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) ↔ 〈“𝑉”〉 ∈ (1 ClWWalksN 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {csn 4630 ‘cfv 6549 (class class class)co 7419 0cc0 11140 1c1 11141 ♯chash 14325 Word cword 14500 〈“cs1 14581 Vtxcvtx 28881 Edgcedg 28932 ClWWalksN cclwwlkn 29906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-xnn0 12578 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-lsw 14549 df-s1 14582 df-clwwlk 29864 df-clwwlkn 29907 |
This theorem is referenced by: (None) |
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