Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwwlknon1 Structured version   Visualization version   GIF version

Theorem clwwlknon1 27266
 Description: The set of closed walks on vertex 𝑋 of length 1 in a graph 𝐺 as words over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.)
Hypotheses
Ref Expression
clwwlknon1.v 𝑉 = (Vtx‘𝐺)
clwwlknon1.c 𝐶 = (ClWWalksNOn‘𝐺)
clwwlknon1.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
clwwlknon1 (𝑋𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
Distinct variable groups:   𝑤,𝐺   𝑤,𝑉   𝑤,𝑋
Allowed substitution hints:   𝐶(𝑤)   𝐸(𝑤)

Proof of Theorem clwwlknon1
StepHypRef Expression
1 clwwlknon1.c . . . 4 𝐶 = (ClWWalksNOn‘𝐺)
21oveqi 6827 . . 3 (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1)
32a1i 11 . 2 (𝑋𝑉 → (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1))
4 clwwlknon 27256 . . 3 (𝑋(ClWWalksNOn‘𝐺)1) = {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
54a1i 11 . 2 (𝑋𝑉 → (𝑋(ClWWalksNOn‘𝐺)1) = {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
6 clwwlkn1 27191 . . . . 5 (𝑤 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
76anbi1i 733 . . . 4 ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
8 clwwlknon1.v . . . . . . . . . . . 12 𝑉 = (Vtx‘𝐺)
98eqcomi 2769 . . . . . . . . . . 11 (Vtx‘𝐺) = 𝑉
109wrdeqi 13534 . . . . . . . . . 10 Word (Vtx‘𝐺) = Word 𝑉
1110eleq2i 2831 . . . . . . . . 9 (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
1211biimpi 206 . . . . . . . 8 (𝑤 ∈ Word (Vtx‘𝐺) → 𝑤 ∈ Word 𝑉)
13123ad2ant2 1129 . . . . . . 7 (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → 𝑤 ∈ Word 𝑉)
1413ad2antrl 766 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 ∈ Word 𝑉)
1513adantr 472 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → 𝑤 ∈ Word 𝑉)
16 simpl1 1228 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (♯‘𝑤) = 1)
17 simpr 479 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
1815, 16, 173jca 1123 . . . . . . . 8 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋))
1918adantl 473 . . . . . . 7 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋))
20 wrdl1s1 13605 . . . . . . . 8 (𝑋𝑉 → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
2120adantr 472 . . . . . . 7 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
2219, 21mpbird 247 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 = ⟨“𝑋”⟩)
23 sneq 4331 . . . . . . . . . . . . 13 ((𝑤‘0) = 𝑋 → {(𝑤‘0)} = {𝑋})
24 clwwlknon1.e . . . . . . . . . . . . . . 15 𝐸 = (Edg‘𝐺)
2524eqcomi 2769 . . . . . . . . . . . . . 14 (Edg‘𝐺) = 𝐸
2625a1i 11 . . . . . . . . . . . . 13 ((𝑤‘0) = 𝑋 → (Edg‘𝐺) = 𝐸)
2723, 26eleq12d 2833 . . . . . . . . . . . 12 ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) ↔ {𝑋} ∈ 𝐸))
2827biimpd 219 . . . . . . . . . . 11 ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) → {𝑋} ∈ 𝐸))
2928a1i 11 . . . . . . . . . 10 (𝑋𝑉 → ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) → {𝑋} ∈ 𝐸)))
3029com13 88 . . . . . . . . 9 ({(𝑤‘0)} ∈ (Edg‘𝐺) → ((𝑤‘0) = 𝑋 → (𝑋𝑉 → {𝑋} ∈ 𝐸)))
31303ad2ant3 1130 . . . . . . . 8 (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → ((𝑤‘0) = 𝑋 → (𝑋𝑉 → {𝑋} ∈ 𝐸)))
3231imp 444 . . . . . . 7 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑋𝑉 → {𝑋} ∈ 𝐸))
3332impcom 445 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → {𝑋} ∈ 𝐸)
3414, 22, 33jca32 559 . . . . 5 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)))
35 fveq2 6353 . . . . . . . . . 10 (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = (♯‘⟨“𝑋”⟩))
36 s1len 13596 . . . . . . . . . 10 (♯‘⟨“𝑋”⟩) = 1
3735, 36syl6eq 2810 . . . . . . . . 9 (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = 1)
3837ad2antrl 766 . . . . . . . 8 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (♯‘𝑤) = 1)
3938adantl 473 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (♯‘𝑤) = 1)
408wrdeqi 13534 . . . . . . . . . 10 Word 𝑉 = Word (Vtx‘𝐺)
4140eleq2i 2831 . . . . . . . . 9 (𝑤 ∈ Word 𝑉𝑤 ∈ Word (Vtx‘𝐺))
4241biimpi 206 . . . . . . . 8 (𝑤 ∈ Word 𝑉𝑤 ∈ Word (Vtx‘𝐺))
4342ad2antrl 766 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → 𝑤 ∈ Word (Vtx‘𝐺))
44 fveq1 6352 . . . . . . . . . . . . . . 15 (𝑤 = ⟨“𝑋”⟩ → (𝑤‘0) = (⟨“𝑋”⟩‘0))
45 s1fv 13601 . . . . . . . . . . . . . . 15 (𝑋𝑉 → (⟨“𝑋”⟩‘0) = 𝑋)
4644, 45sylan9eq 2814 . . . . . . . . . . . . . 14 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → (𝑤‘0) = 𝑋)
4746eqcomd 2766 . . . . . . . . . . . . 13 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → 𝑋 = (𝑤‘0))
4847sneqd 4333 . . . . . . . . . . . 12 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → {𝑋} = {(𝑤‘0)})
4924a1i 11 . . . . . . . . . . . 12 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → 𝐸 = (Edg‘𝐺))
5048, 49eleq12d 2833 . . . . . . . . . . 11 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → ({𝑋} ∈ 𝐸 ↔ {(𝑤‘0)} ∈ (Edg‘𝐺)))
5150biimpd 219 . . . . . . . . . 10 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → ({𝑋} ∈ 𝐸 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5251impancom 455 . . . . . . . . 9 ((𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸) → (𝑋𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5352adantl 473 . . . . . . . 8 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5453impcom 445 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → {(𝑤‘0)} ∈ (Edg‘𝐺))
5539, 43, 543jca 1123 . . . . . 6 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
5646ex 449 . . . . . . . 8 (𝑤 = ⟨“𝑋”⟩ → (𝑋𝑉 → (𝑤‘0) = 𝑋))
5756ad2antrl 766 . . . . . . 7 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋𝑉 → (𝑤‘0) = 𝑋))
5857impcom 445 . . . . . 6 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (𝑤‘0) = 𝑋)
5955, 58jca 555 . . . . 5 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
6034, 59impbida 913 . . . 4 (𝑋𝑉 → ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
617, 60syl5bb 272 . . 3 (𝑋𝑉 → ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
6261rabbidva2 3326 . 2 (𝑋𝑉 → {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
633, 5, 623eqtrd 2798 1 (𝑋𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  {crab 3054  {csn 4321  ‘cfv 6049  (class class class)co 6814  0cc0 10148  1c1 10149  ♯chash 13331  Word cword 13497  ⟨“cs1 13500  Vtxcvtx 26094  Edgcedg 26159   ClWWalksN cclwwlkn 27168  ClWWalksNOncclwwlknon 27253 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-xnn0 11576  df-z 11590  df-uz 11900  df-fz 12540  df-fzo 12680  df-hash 13332  df-word 13505  df-lsw 13506  df-s1 13508  df-clwwlk 27126  df-clwwlkn 27170  df-clwwlknon 27254 This theorem is referenced by:  clwwlknon1loop  27267  clwwlknon1nloop  27268
 Copyright terms: Public domain W3C validator