MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uspgredg2v Structured version   Visualization version   GIF version

Theorem uspgredg2v 26337
Description: In a simple pseudograph, the mapping of edges having a fixed endpoint to the "other" vertex of the edge (which may be the fixed vertex itself in the case of a loop) is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.)
Hypotheses
Ref Expression
uspgredg2v.v 𝑉 = (Vtx‘𝐺)
uspgredg2v.e 𝐸 = (Edg‘𝐺)
uspgredg2v.a 𝐴 = {𝑒𝐸𝑁𝑒}
uspgredg2v.f 𝐹 = (𝑦𝐴 ↦ (𝑧𝑉 𝑦 = {𝑁, 𝑧}))
Assertion
Ref Expression
uspgredg2v ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
Distinct variable groups:   𝑒,𝐸   𝑧,𝐺   𝑒,𝑁   𝑧,𝑁   𝑧,𝑉   𝑦,𝐴   𝑦,𝐺   𝑦,𝑁,𝑧   𝑦,𝑉   𝑦,𝑒
Allowed substitution hints:   𝐴(𝑧,𝑒)   𝐸(𝑦,𝑧)   𝐹(𝑦,𝑧,𝑒)   𝐺(𝑒)   𝑉(𝑒)

Proof of Theorem uspgredg2v
Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uspgredg2v.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 uspgredg2v.e . . . . 5 𝐸 = (Edg‘𝐺)
3 uspgredg2v.a . . . . 5 𝐴 = {𝑒𝐸𝑁𝑒}
41, 2, 3uspgredg2vlem 26336 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑦𝐴) → (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
54ralrimiva 3114 . . 3 (𝐺 ∈ USPGraph → ∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
65adantr 466 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → ∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
7 preq2 4403 . . . . . . 7 (𝑧 = 𝑛 → {𝑁, 𝑧} = {𝑁, 𝑛})
87eqeq2d 2780 . . . . . 6 (𝑧 = 𝑛 → (𝑦 = {𝑁, 𝑧} ↔ 𝑦 = {𝑁, 𝑛}))
98cbvriotav 6764 . . . . 5 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑛𝑉 𝑦 = {𝑁, 𝑛})
107eqeq2d 2780 . . . . . 6 (𝑧 = 𝑛 → (𝑥 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑛}))
1110cbvriotav 6764 . . . . 5 (𝑧𝑉 𝑥 = {𝑁, 𝑧}) = (𝑛𝑉 𝑥 = {𝑁, 𝑛})
12 simpl 468 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐺 ∈ USPGraph)
13 eleq2w 2833 . . . . . . . . . . 11 (𝑒 = 𝑦 → (𝑁𝑒𝑁𝑦))
1413, 3elrab2 3516 . . . . . . . . . 10 (𝑦𝐴 ↔ (𝑦𝐸𝑁𝑦))
152eleq2i 2841 . . . . . . . . . . . 12 (𝑦𝐸𝑦 ∈ (Edg‘𝐺))
1615biimpi 206 . . . . . . . . . . 11 (𝑦𝐸𝑦 ∈ (Edg‘𝐺))
1716anim1i 594 . . . . . . . . . 10 ((𝑦𝐸𝑁𝑦) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
1814, 17sylbi 207 . . . . . . . . 9 (𝑦𝐴 → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
1918adantr 466 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
2012, 19anim12i 592 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦)))
21 3anass 1079 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) ↔ (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦)))
2220, 21sylibr 224 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
23 uspgredg2vtxeu 26333 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
24 reueq1 3288 . . . . . . . 8 (𝑉 = (Vtx‘𝐺) → (∃!𝑛𝑉 𝑦 = {𝑁, 𝑛} ↔ ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛}))
251, 24ax-mp 5 . . . . . . 7 (∃!𝑛𝑉 𝑦 = {𝑁, 𝑛} ↔ ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
2623, 25sylibr 224 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) → ∃!𝑛𝑉 𝑦 = {𝑁, 𝑛})
2722, 26syl 17 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → ∃!𝑛𝑉 𝑦 = {𝑁, 𝑛})
28 eleq2w 2833 . . . . . . . . . . 11 (𝑒 = 𝑥 → (𝑁𝑒𝑁𝑥))
2928, 3elrab2 3516 . . . . . . . . . 10 (𝑥𝐴 ↔ (𝑥𝐸𝑁𝑥))
302eleq2i 2841 . . . . . . . . . . . 12 (𝑥𝐸𝑥 ∈ (Edg‘𝐺))
3130biimpi 206 . . . . . . . . . . 11 (𝑥𝐸𝑥 ∈ (Edg‘𝐺))
3231anim1i 594 . . . . . . . . . 10 ((𝑥𝐸𝑁𝑥) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3329, 32sylbi 207 . . . . . . . . 9 (𝑥𝐴 → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3433adantl 467 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3512, 34anim12i 592 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥)))
36 3anass 1079 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥) ↔ (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥)))
3735, 36sylibr 224 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
38 uspgredg2vtxeu 26333 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
39 reueq1 3288 . . . . . . . 8 (𝑉 = (Vtx‘𝐺) → (∃!𝑛𝑉 𝑥 = {𝑁, 𝑛} ↔ ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛}))
401, 39ax-mp 5 . . . . . . 7 (∃!𝑛𝑉 𝑥 = {𝑁, 𝑛} ↔ ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
4138, 40sylibr 224 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥) → ∃!𝑛𝑉 𝑥 = {𝑁, 𝑛})
4237, 41syl 17 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → ∃!𝑛𝑉 𝑥 = {𝑁, 𝑛})
439, 11, 27, 42riotaeqimp 6776 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) ∧ (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧})) → 𝑦 = 𝑥)
4443ex 397 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
4544ralrimivva 3119 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → ∀𝑦𝐴𝑥𝐴 ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
46 uspgredg2v.f . . 3 𝐹 = (𝑦𝐴 ↦ (𝑧𝑉 𝑦 = {𝑁, 𝑧}))
47 eqeq1 2774 . . . 4 (𝑦 = 𝑥 → (𝑦 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑧}))
4847riotabidv 6755 . . 3 (𝑦 = 𝑥 → (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}))
4946, 48f1mpt 6660 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉 ∧ ∀𝑦𝐴𝑥𝐴 ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥)))
506, 45, 49sylanbrc 564 1 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  ∃!wreu 3062  {crab 3064  {cpr 4316  cmpt 4861  1-1wf1 6028  cfv 6031  crio 6752  Vtxcvtx 26094  Edgcedg 26159  USPGraphcuspgr 26264
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-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-rmo 3068  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-2o 7713  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-cda 9191  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-xnn0 11565  df-z 11579  df-uz 11888  df-fz 12533  df-hash 13321  df-edg 26160  df-upgr 26197  df-uspgr 26266
This theorem is referenced by:  uspgredgleord  26346
  Copyright terms: Public domain W3C validator