Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1pthon2v-av Structured version   Visualization version   GIF version

Theorem 1pthon2v-av 41320
 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-av.v 𝑉 = (Vtx‘𝐺)
1pthon2v-av.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
1pthon2v-av ((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-av
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 simpl 472 . . . . . . . 8 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
21anim2i 591 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉)) → (𝐺 ∈ UHGraph ∧ 𝐴𝑉))
323adant3 1074 . . . . . 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-av.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
650pthonv-av 41297 . . . . . 6 (𝐴𝑉 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝)
76adantl 481 . . . . 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 6557 . . . . . . . 8 (𝐵 = 𝐴 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
109eqcoms 2618 . . . . . . 7 (𝐴 = 𝐵 → (𝐴(PathsOn‘𝐺)𝐵) = (𝐴(PathsOn‘𝐺)𝐴))
1110breqd 4594 . . . . . 6 (𝐴 = 𝐵 → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
12112exbidv 1839 . . . . 5 (𝐴 = 𝐵 → (∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐴)𝑝))
1312adantr 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))
148, 13mpbird 246 . . 3 ((𝐴 = 𝐵 ∧ (𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
1514ex 449 . 2 (𝐴 = 𝐵 → ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
16 1pthon2v-av.e . . . . . . . . . . 11 𝐸 = (Edg‘𝐺)
1716eleq2i 2680 . . . . . . . . . 10 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
18 eqid 2610 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
1918uhgredgiedgb 25799 . . . . . . . . . 10 (𝐺 ∈ UHGraph → (𝑒 ∈ (Edg‘𝐺) ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
2017, 19syl5bb 271 . . . . . . . . 9 (𝐺 ∈ UHGraph → (𝑒𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
21203ad2ant1 1075 . . . . . . . 8 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑒𝐸 ↔ ∃𝑖 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑖)))
22 s1cli 13237 . . . . . . . . . . . 12 ⟨“𝑖”⟩ ∈ Word V
23 s2cli 13475 . . . . . . . . . . . 12 ⟨“𝐴𝐵”⟩ ∈ Word V
2422, 23pm3.2i 470 . . . . . . . . . . 11 (⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V)
25 eqid 2610 . . . . . . . . . . . 12 ⟨“𝐴𝐵”⟩ = ⟨“𝐴𝐵”⟩
26 eqid 2610 . . . . . . . . . . . 12 ⟨“𝑖”⟩ = ⟨“𝑖”⟩
27 simpl2l 1107 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐴𝑉)
28 simpl2r 1108 . . . . . . . . . . . 12 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → 𝐵𝑉)
29 eqneqall 2793 . . . . . . . . . . . . . . . 16 (𝐴 = 𝐵 → (𝐴𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3029com12 32 . . . . . . . . . . . . . . 15 (𝐴𝐵 → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
31303ad2ant3 1077 . . . . . . . . . . . . . 14 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3231adantr 480 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → (𝐴 = 𝐵 → ((iEdg‘𝐺)‘𝑖) = {𝐴}))
3332imp 444 . . . . . . . . . . . 12 ((((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) ∧ 𝐴 = 𝐵) → ((iEdg‘𝐺)‘𝑖) = {𝐴})
34 sseq2 3590 . . . . . . . . . . . . . . . 16 (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)))
3534adantl 481 . . . . . . . . . . . . . . 15 ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) → ({𝐴, 𝐵} ⊆ 𝑒 ↔ {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖)))
3635biimpa 500 . . . . . . . . . . . . . 14 (((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
3736adantl 481 . . . . . . . . . . . . 13 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → {𝐴, 𝐵} ⊆ ((iEdg‘𝐺)‘𝑖))
3837adantr 480 . . . . . . . . . . . 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 41311 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩)
40 breq12 4588 . . . . . . . . . . . 12 ((𝑓 = ⟨“𝑖”⟩ ∧ 𝑝 = ⟨“𝐴𝐵”⟩) → (𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝 ↔ ⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩))
4140spc2egv 3268 . . . . . . . . . . 11 ((⟨“𝑖”⟩ ∈ Word V ∧ ⟨“𝐴𝐵”⟩ ∈ Word V) → (⟨“𝑖”⟩(𝐴(PathsOn‘𝐺)𝐵)⟨“𝐴𝐵”⟩ → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
4224, 39, 41mpsyl 66 . . . . . . . . . 10 (((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝑖 ∈ dom (iEdg‘𝐺) ∧ 𝑒 = ((iEdg‘𝐺)‘𝑖)) ∧ {𝐴, 𝐵} ⊆ 𝑒)) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
4342exp44 639 . . . . . . . . 9 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑖 ∈ dom (iEdg‘𝐺) → (𝑒 = ((iEdg‘𝐺)‘𝑖) → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
4443rexlimdv 3012 . . . . . . . 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 229 . . . . . . 7 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑒𝐸 → ({𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)))
4645rexlimdv 3012 . . . . . 6 ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))
47463exp 1256 . . . . 5 (𝐺 ∈ UHGraph → ((𝐴𝑉𝐵𝑉) → (𝐴𝐵 → (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
4847com34 89 . . . 4 (𝐺 ∈ UHGraph → ((𝐴𝑉𝐵𝑉) → (∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒 → (𝐴𝐵 → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝))))
49483imp 1249 . . 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 2865 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  Word cword 13146  ⟨“cs1 13149  ⟨“cs2 13437  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722  Edgcedga 25792  PathsOncpthson 40921 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-uhgr 25724  df-edga 25793  df-1wlks 40800  df-wlkson 40802  df-trls 40901  df-trlson 40902  df-pths 40923  df-pthson 40925 This theorem is referenced by:  1pthon2ve  41321  cusconngr  41358
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