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Theorem 1pthon2v 26896
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 473 . . . . . . . 8 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
21anim2i 592 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉)) → (𝐺 ∈ UHGraph ∧ 𝐴𝑉))
323adant3 1079 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → (𝐺 ∈ UHGraph ∧ 𝐴𝑉))
43adantl 482 . . . . 5 ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → (𝐺 ∈ UHGraph ∧ 𝐴𝑉))
5 1pthon2v.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
650pthonv 26873 . . . . . 6 (𝐴𝑉 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)
76adantl 482 . . . . 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 6618 . . . . . . . 8 (𝐵 = 𝐴 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
109eqcoms 2629 . . . . . . 7 (𝐴 = 𝐵 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
1110breqd 4629 . . . . . 6 (𝐴 = 𝐵 → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
12112exbidv 1849 . . . . 5 (𝐴 = 𝐵 → (∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
1312adantr 481 . . . 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 450 . 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 2690 . . . . . . . . . 10 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
18 eqid 2621 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
1918uhgredgiedgb 25933 . . . . . . . . . 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 1080 . . . . . . . 8 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑒𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
22 s1cli 13331 . . . . . . . . . . . 12 ⟨“𝑖”⟩ ∈ Word V
23 s2cli 13569 . . . . . . . . . . . 12 ⟨“𝐴𝐵”⟩ ∈ Word V
2422, 23pm3.2i 471 . . . . . . . . . . 11 (⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V)
25 eqid 2621 . . . . . . . . . . . 12 ⟨“𝐴𝐵”⟩ = ⟨“𝐴𝐵”⟩
26 eqid 2621 . . . . . . . . . . . 12 ⟨“𝑖”⟩ = ⟨“𝑖”⟩
27 simpl2l 1112 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐴𝑉)
28 simpl2r 1113 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐵𝑉)
29 eqneqall 2801 . . . . . . . . . . . . . . . 16 (𝐴 = 𝐵 → (𝐴𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3029com12 32 . . . . . . . . . . . . . . 15 (𝐴𝐵 → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
31303ad2ant3 1082 . . . . . . . . . . . . . 14 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3231adantr 481 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3332imp 445 . . . . . . . . . . . 12 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) ∧ 𝐴 = 𝐵) → ((iEdg‘𝐺)‘𝑖) = {𝐴})
34 sseq2 3611 . . . . . . . . . . . . . . . 16 (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)))
3534adantl 482 . . . . . . . . . . . . . . 15 ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)))
3635biimpa 501 . . . . . . . . . . . . . 14 (((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
3736adantl 482 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
3837adantr 481 . . . . . . . . . . . 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 26887 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩)
40 breq12 4623 . . . . . . . . . . . 12 ((𝑓 = ⟨“𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵”⟩) → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩))
4140spc2egv 3284 . . . . . . . . . . 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 640 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑖 ∈ dom (iEdg‘𝐺) → (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
4443rexlimdv 3024 . . . . . . . 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 3024 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
47463exp 1261 . . . . 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 1254 . . 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 2873 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 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  Vcvv 3189  wss 3559  {csn 4153  {cpr 4155   class class class wbr 4618  dom cdm 5079  cfv 5852  (class class class)co 6610  Word cword 13238  ⟨“cs1 13241  ⟨“cs2 13531  Vtxcvtx 25791  iEdgciedg 25792  Edgcedg 25856   UHGraph cuhgr 25864  PathsOncpthson 26496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-hash 13066  df-word 13246  df-concat 13248  df-s1 13249  df-s2 13538  df-edg 25857  df-uhgr 25866  df-wlks 26382  df-wlkson 26383  df-trls 26475  df-trlson 26476  df-pths 26498  df-pthson 26500
This theorem is referenced by:  1pthon2ve  26897  cusconngr  26934
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