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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 27819 | . 2 ⊢ (〈“𝑉”〉 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺))) | |
2 | s1fv 13964 | . . . . . . . 8 ⊢ (𝑉 ∈ (Vtx‘𝐺) → (〈“𝑉”〉‘0) = 𝑉) | |
3 | 2 | sneqd 4579 | . . . . . . 7 ⊢ (𝑉 ∈ (Vtx‘𝐺) → {(〈“𝑉”〉‘0)} = {𝑉}) |
4 | 3 | eleq1d 2897 | . . . . . 6 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺) ↔ {𝑉} ∈ (Edg‘𝐺))) |
5 | 4 | biimpcd 251 | . . . . 5 ⊢ ({(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺) → (𝑉 ∈ (Vtx‘𝐺) → {𝑉} ∈ (Edg‘𝐺))) |
6 | 5 | 3ad2ant3 1131 | . . . 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 13960 | . . . . . 6 ⊢ (♯‘〈“𝑉”〉) = 1 | |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → (♯‘〈“𝑉”〉) = 1) |
10 | s1cl 13956 | . . . . . 6 ⊢ (𝑉 ∈ (Vtx‘𝐺) → 〈“𝑉”〉 ∈ Word (Vtx‘𝐺)) | |
11 | 10 | adantr 483 | . . . . 5 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → 〈“𝑉”〉 ∈ Word (Vtx‘𝐺)) |
12 | 2 | eqcomd 2827 | . . . . . . . 8 ⊢ (𝑉 ∈ (Vtx‘𝐺) → 𝑉 = (〈“𝑉”〉‘0)) |
13 | 12 | sneqd 4579 | . . . . . . 7 ⊢ (𝑉 ∈ (Vtx‘𝐺) → {𝑉} = {(〈“𝑉”〉‘0)}) |
14 | 13 | eleq1d 2897 | . . . . . 6 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) ↔ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺))) |
15 | 14 | biimpa 479 | . . . . 5 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)) |
16 | 9, 11, 15 | 3jca 1124 | . . . 4 ⊢ ((𝑉 ∈ (Vtx‘𝐺) ∧ {𝑉} ∈ (Edg‘𝐺)) → ((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺))) |
17 | 16 | ex 415 | . . 3 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) → ((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)))) |
18 | 7, 17 | impbid 214 | . 2 ⊢ (𝑉 ∈ (Vtx‘𝐺) → (((♯‘〈“𝑉”〉) = 1 ∧ 〈“𝑉”〉 ∈ Word (Vtx‘𝐺) ∧ {(〈“𝑉”〉‘0)} ∈ (Edg‘𝐺)) ↔ {𝑉} ∈ (Edg‘𝐺))) |
19 | 1, 18 | syl5rbb 286 | 1 ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) ↔ 〈“𝑉”〉 ∈ (1 ClWWalksN 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {csn 4567 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 ♯chash 13691 Word cword 13862 〈“cs1 13949 Vtxcvtx 26781 Edgcedg 26832 ClWWalksN cclwwlkn 27802 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-lsw 13915 df-s1 13950 df-clwwlk 27760 df-clwwlkn 27803 |
This theorem is referenced by: (None) |
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