MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwwlkn1loopb Structured version   Visualization version   GIF version

Theorem clwwlkn1loopb 29560
Description: A word represents a closed walk of length 1 iff this word is a singleton word consisting of a vertex with an attached loop. (Contributed by AV, 11-Feb-2022.)
Assertion
Ref Expression
clwwlkn1loopb (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
Distinct variable groups:   𝑣,𝐺   𝑣,𝑊

Proof of Theorem clwwlkn1loopb
StepHypRef Expression
1 clwwlkn1 29558 . 2 (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
2 wrdl1exs1 14568 . . . . . 6 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ∃𝑣 ∈ (Vtx‘𝐺)𝑊 = ⟨“𝑣”⟩)
3 fveq1 6891 . . . . . . . . . . . . . . 15 (𝑊 = ⟨“𝑣”⟩ → (𝑊‘0) = (⟨“𝑣”⟩‘0))
4 s1fv 14565 . . . . . . . . . . . . . . 15 (𝑣 ∈ (Vtx‘𝐺) → (⟨“𝑣”⟩‘0) = 𝑣)
53, 4sylan9eq 2791 . . . . . . . . . . . . . 14 ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊‘0) = 𝑣)
65sneqd 4641 . . . . . . . . . . . . 13 ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → {(𝑊‘0)} = {𝑣})
76eleq1d 2817 . . . . . . . . . . . 12 ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → ({(𝑊‘0)} ∈ (Edg‘𝐺) ↔ {𝑣} ∈ (Edg‘𝐺)))
87biimpd 228 . . . . . . . . . . 11 ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → {𝑣} ∈ (Edg‘𝐺)))
98ex 412 . . . . . . . . . 10 (𝑊 = ⟨“𝑣”⟩ → (𝑣 ∈ (Vtx‘𝐺) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → {𝑣} ∈ (Edg‘𝐺))))
109com13 88 . . . . . . . . 9 ({(𝑊‘0)} ∈ (Edg‘𝐺) → (𝑣 ∈ (Vtx‘𝐺) → (𝑊 = ⟨“𝑣”⟩ → {𝑣} ∈ (Edg‘𝐺))))
1110imp 406 . . . . . . . 8 (({(𝑊‘0)} ∈ (Edg‘𝐺) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝑣”⟩ → {𝑣} ∈ (Edg‘𝐺)))
1211ancld 550 . . . . . . 7 (({(𝑊‘0)} ∈ (Edg‘𝐺) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝑣”⟩ → (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
1312reximdva 3167 . . . . . 6 ({(𝑊‘0)} ∈ (Edg‘𝐺) → (∃𝑣 ∈ (Vtx‘𝐺)𝑊 = ⟨“𝑣”⟩ → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
142, 13syl5com 31 . . . . 5 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
1514expcom 413 . . . 4 ((♯‘𝑊) = 1 → (𝑊 ∈ Word (Vtx‘𝐺) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))))
16153imp 1110 . . 3 (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
17 s1len 14561 . . . . . . . 8 (♯‘⟨“𝑣”⟩) = 1
1817a1i 11 . . . . . . 7 ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → (♯‘⟨“𝑣”⟩) = 1)
19 s1cl 14557 . . . . . . . 8 (𝑣 ∈ (Vtx‘𝐺) → ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺))
2019adantr 480 . . . . . . 7 ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺))
214eqcomd 2737 . . . . . . . . . . 11 (𝑣 ∈ (Vtx‘𝐺) → 𝑣 = (⟨“𝑣”⟩‘0))
2221sneqd 4641 . . . . . . . . . 10 (𝑣 ∈ (Vtx‘𝐺) → {𝑣} = {(⟨“𝑣”⟩‘0)})
2322eleq1d 2817 . . . . . . . . 9 (𝑣 ∈ (Vtx‘𝐺) → ({𝑣} ∈ (Edg‘𝐺) ↔ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
2423biimpd 228 . . . . . . . 8 (𝑣 ∈ (Vtx‘𝐺) → ({𝑣} ∈ (Edg‘𝐺) → {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
2524imp 406 . . . . . . 7 ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺))
2618, 20, 253jca 1127 . . . . . 6 ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
2726adantrl 713 . . . . 5 ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
28 fveqeq2 6901 . . . . . . 7 (𝑊 = ⟨“𝑣”⟩ → ((♯‘𝑊) = 1 ↔ (♯‘⟨“𝑣”⟩) = 1))
29 eleq1 2820 . . . . . . 7 (𝑊 = ⟨“𝑣”⟩ → (𝑊 ∈ Word (Vtx‘𝐺) ↔ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺)))
303sneqd 4641 . . . . . . . 8 (𝑊 = ⟨“𝑣”⟩ → {(𝑊‘0)} = {(⟨“𝑣”⟩‘0)})
3130eleq1d 2817 . . . . . . 7 (𝑊 = ⟨“𝑣”⟩ → ({(𝑊‘0)} ∈ (Edg‘𝐺) ↔ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
3228, 29, 313anbi123d 1435 . . . . . 6 (𝑊 = ⟨“𝑣”⟩ → (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺))))
3332ad2antrl 725 . . . . 5 ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺))))
3427, 33mpbird 256 . . . 4 ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
3534rexlimiva 3146 . . 3 (∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)) → ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
3616, 35impbii 208 . 2 (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
371, 36bitri 274 1 (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wrex 3069  {csn 4629  cfv 6544  (class class class)co 7412  0cc0 11113  1c1 11114  chash 14295  Word cword 14469  ⟨“cs1 14550  Vtxcvtx 28520  Edgcedg 28571   ClWWalksN cclwwlkn 29541
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-lsw 14518  df-s1 14551  df-clwwlk 29499  df-clwwlkn 29542
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator