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Theorem 1pthon2v 30172
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 482 . . . . . . . 8 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
21anim2i 617 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉)) → (𝐺 ∈ UHGraph ∧ 𝐴𝑉))
323adant3 1133 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → (𝐺 ∈ UHGraph ∧ 𝐴𝑉))
43adantl 481 . . . . 5 ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → (𝐺 ∈ UHGraph ∧ 𝐴𝑉))
5 1pthon2v.v . . . . . 6 𝑉 = (Vtx‘𝐺)
650pthonv 30148 . . . . 5 (𝐴𝑉 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)
74, 6simpl2im 503 . . . 4 ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)
8 oveq2 7439 . . . . . . . 8 (𝐵 = 𝐴 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
98eqcoms 2745 . . . . . . 7 (𝐴 = 𝐵 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
109breqd 5154 . . . . . 6 (𝐴 = 𝐵 → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
11102exbidv 1924 . . . . 5 (𝐴 = 𝐵 → (∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
1211adantr 480 . . . 4 ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → (∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
137, 12mpbird 257 . . 3 ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
1413ex 412 . 2 (𝐴 = 𝐵 → ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
15 1pthon2v.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
1615eleq2i 2833 . . . . . . . . . 10 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
17 eqid 2737 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
1817uhgredgiedgb 29143 . . . . . . . . . 10 (𝐺 ∈ UHGraph → (𝑒 ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
1916, 18bitrid 283 . . . . . . . . 9 (𝐺 ∈ UHGraph → (𝑒𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
20193ad2ant1 1134 . . . . . . . 8 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑒𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
21 s1cli 14643 . . . . . . . . . . . 12 ⟨“𝑖”⟩ ∈ Word V
22 s2cli 14919 . . . . . . . . . . . 12 ⟨“𝐴𝐵”⟩ ∈ Word V
2321, 22pm3.2i 470 . . . . . . . . . . 11 (⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V)
24 eqid 2737 . . . . . . . . . . . 12 ⟨“𝐴𝐵”⟩ = ⟨“𝐴𝐵”⟩
25 eqid 2737 . . . . . . . . . . . 12 ⟨“𝑖”⟩ = ⟨“𝑖”⟩
26 simpl2l 1227 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐴𝑉)
27 simpl2r 1228 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐵𝑉)
28 eqneqall 2951 . . . . . . . . . . . . . . . 16 (𝐴 = 𝐵 → (𝐴𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
2928com12 32 . . . . . . . . . . . . . . 15 (𝐴𝐵 → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
30293ad2ant3 1136 . . . . . . . . . . . . . 14 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3130adantr 480 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3231imp 406 . . . . . . . . . . . 12 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) ∧ 𝐴 = 𝐵) → ((iEdg‘𝐺)‘𝑖) = {𝐴})
33 sseq2 4010 . . . . . . . . . . . . . . . 16 (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)))
3433adantl 481 . . . . . . . . . . . . . . 15 ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)))
3534biimpa 476 . . . . . . . . . . . . . 14 (((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
3635adantl 481 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
3736adantr 480 . . . . . . . . . . . 12 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) ∧ 𝐴𝐵) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
3824, 25, 26, 27, 32, 37, 5, 171pthond 30163 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩)
39 breq12 5148 . . . . . . . . . . . 12 ((𝑓 = ⟨“𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵”⟩) → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩))
4039spc2egv 3599 . . . . . . . . . . 11 ((⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V) → (⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩ → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
4123, 38, 40mpsyl 68 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
4241exp44 437 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑖 ∈ dom (iEdg‘𝐺) → (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
4342rexlimdv 3153 . . . . . . . 8 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)))
4420, 43sylbid 240 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑒𝐸 → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)))
4544rexlimdv 3153 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
46453exp 1120 . . . . 5 (𝐺 ∈ UHGraph → ((𝐴𝑉𝐵𝑉) → (𝐴𝐵 → (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
4746com34 91 . . . 4 (𝐺 ∈ UHGraph → ((𝐴𝑉𝐵𝑉) → (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 → (𝐴𝐵 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
48473imp 1111 . . 3 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → (𝐴𝐵 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
4948com12 32 . 2 (𝐴𝐵 → ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
5014, 49pm2.61ine 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 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wrex 3070  Vcvv 3480  wss 3951  {csn 4626  {cpr 4628   class class class wbr 5143  dom cdm 5685  cfv 6561  (class class class)co 7431  Word cword 14552  ⟨“cs1 14633  ⟨“cs2 14880  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UHGraphcuhgr 29073  PathsOncpthson 29732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-edg 29065  df-uhgr 29075  df-wlks 29617  df-wlkson 29618  df-trls 29710  df-trlson 29711  df-pths 29734  df-pthson 29736
This theorem is referenced by:  1pthon2ve  30173  cusconngr  30210
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