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Theorem 1pthon2v 27330
Description: For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
Hypotheses
Ref Expression
1pthon2v.v 𝑉 = (Vtx‘𝐺)
1pthon2v.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
1pthon2v ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
Distinct variable groups:   𝐴,𝑒,𝑓,𝑝   𝐵,𝑒,𝑓,𝑝   𝑒,𝐺,𝑓,𝑝   𝑒,𝑉
Allowed substitution hints:   𝐸(𝑒,𝑓,𝑝)   𝑉(𝑓,𝑝)

Proof of Theorem 1pthon2v
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 simpl 468 . . . . . . . 8 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
21anim2i 595 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉)) → (𝐺 ∈ UHGraph ∧ 𝐴𝑉))
323adant3 1125 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → (𝐺 ∈ UHGraph ∧ 𝐴𝑉))
43adantl 467 . . . . 5 ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → (𝐺 ∈ UHGraph ∧ 𝐴𝑉))
5 1pthon2v.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
650pthonv 27306 . . . . . 6 (𝐴𝑉 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)
76adantl 467 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝐴𝑉) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)
84, 7syl 17 . . . 4 ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)
9 oveq2 6800 . . . . . . . 8 (𝐵 = 𝐴 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
109eqcoms 2778 . . . . . . 7 (𝐴 = 𝐵 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
1110breqd 4795 . . . . . 6 (𝐴 = 𝐵 → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
12112exbidv 2003 . . . . 5 (𝐴 = 𝐵 → (∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
1312adantr 466 . . . 4 ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → (∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
148, 13mpbird 247 . . 3 ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
1514ex 397 . 2 (𝐴 = 𝐵 → ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
16 1pthon2v.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
1716eleq2i 2841 . . . . . . . . . 10 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
18 eqid 2770 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
1918uhgredgiedgb 26241 . . . . . . . . . 10 (𝐺 ∈ UHGraph → (𝑒 ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
2017, 19syl5bb 272 . . . . . . . . 9 (𝐺 ∈ UHGraph → (𝑒𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
21203ad2ant1 1126 . . . . . . . 8 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑒𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
22 s1cli 13584 . . . . . . . . . . . 12 ⟨“𝑖”⟩ ∈ Word V
23 s2cli 13833 . . . . . . . . . . . 12 ⟨“𝐴𝐵”⟩ ∈ Word V
2422, 23pm3.2i 447 . . . . . . . . . . 11 (⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V)
25 eqid 2770 . . . . . . . . . . . 12 ⟨“𝐴𝐵”⟩ = ⟨“𝐴𝐵”⟩
26 eqid 2770 . . . . . . . . . . . 12 ⟨“𝑖”⟩ = ⟨“𝑖”⟩
27 simpl2l 1281 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐴𝑉)
28 simpl2r 1283 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐵𝑉)
29 eqneqall 2953 . . . . . . . . . . . . . . . 16 (𝐴 = 𝐵 → (𝐴𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3029com12 32 . . . . . . . . . . . . . . 15 (𝐴𝐵 → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
31303ad2ant3 1128 . . . . . . . . . . . . . 14 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3231adantr 466 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3332imp 393 . . . . . . . . . . . 12 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) ∧ 𝐴 = 𝐵) → ((iEdg‘𝐺)‘𝑖) = {𝐴})
34 sseq2 3774 . . . . . . . . . . . . . . . 16 (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)))
3534adantl 467 . . . . . . . . . . . . . . 15 ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)))
3635biimpa 462 . . . . . . . . . . . . . 14 (((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
3736adantl 467 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
3837adantr 466 . . . . . . . . . . . 12 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) ∧ 𝐴𝐵) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
3925, 26, 27, 28, 33, 38, 5, 181pthond 27321 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩)
40 breq12 4789 . . . . . . . . . . . 12 ((𝑓 = ⟨“𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵”⟩) → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩))
4140spc2egv 3444 . . . . . . . . . . 11 ((⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V) → (⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩ → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
4224, 39, 41mpsyl 68 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
4342exp44 424 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑖 ∈ dom (iEdg‘𝐺) → (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
4443rexlimdv 3177 . . . . . . . 8 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)))
4521, 44sylbid 230 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑒𝐸 → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)))
4645rexlimdv 3177 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
47463exp 1111 . . . . 5 (𝐺 ∈ UHGraph → ((𝐴𝑉𝐵𝑉) → (𝐴𝐵 → (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
4847com34 91 . . . 4 (𝐺 ∈ UHGraph → ((𝐴𝑉𝐵𝑉) → (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 → (𝐴𝐵 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
49483imp 1100 . . 3 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → (𝐴𝐵 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
5049com12 32 . 2 (𝐴𝐵 → ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
5115, 50pm2.61ine 3025 1 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  wne 2942  wrex 3061  Vcvv 3349  wss 3721  {csn 4314  {cpr 4316   class class class wbr 4784  dom cdm 5249  cfv 6031  (class class class)co 6792  Word cword 13486  ⟨“cs1 13489  ⟨“cs2 13794  Vtxcvtx 26094  iEdgciedg 26095  Edgcedg 26159  UHGraphcuhgr 26171  PathsOncpthson 26844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-ifp 1049  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-n0 11494  df-z 11579  df-uz 11888  df-fz 12533  df-fzo 12673  df-hash 13321  df-word 13494  df-concat 13496  df-s1 13497  df-s2 13801  df-edg 26160  df-uhgr 26173  df-wlks 26729  df-wlkson 26730  df-trls 26823  df-trlson 26824  df-pths 26846  df-pthson 26848
This theorem is referenced by:  1pthon2ve  27331  cusconngr  27368
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