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Theorem uspgredg2v 27098
 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 27097 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑦𝐴) → (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
54ralrimiva 3111 . . 3 (𝐺 ∈ USPGraph → ∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
65adantr 485 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → ∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
7 preq2 4620 . . . . . . 7 (𝑧 = 𝑛 → {𝑁, 𝑧} = {𝑁, 𝑛})
87eqeq2d 2770 . . . . . 6 (𝑧 = 𝑛 → (𝑦 = {𝑁, 𝑧} ↔ 𝑦 = {𝑁, 𝑛}))
98cbvriotavw 7111 . . . . 5 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑛𝑉 𝑦 = {𝑁, 𝑛})
107eqeq2d 2770 . . . . . 6 (𝑧 = 𝑛 → (𝑥 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑛}))
1110cbvriotavw 7111 . . . . 5 (𝑧𝑉 𝑥 = {𝑁, 𝑧}) = (𝑛𝑉 𝑥 = {𝑁, 𝑛})
12 simpl 487 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐺 ∈ USPGraph)
13 eleq2w 2834 . . . . . . . . . . 11 (𝑒 = 𝑦 → (𝑁𝑒𝑁𝑦))
1413, 3elrab2 3603 . . . . . . . . . 10 (𝑦𝐴 ↔ (𝑦𝐸𝑁𝑦))
152eleq2i 2842 . . . . . . . . . . . 12 (𝑦𝐸𝑦 ∈ (Edg‘𝐺))
1615biimpi 219 . . . . . . . . . . 11 (𝑦𝐸𝑦 ∈ (Edg‘𝐺))
1716anim1i 618 . . . . . . . . . 10 ((𝑦𝐸𝑁𝑦) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
1814, 17sylbi 220 . . . . . . . . 9 (𝑦𝐴 → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
1918adantr 485 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
2012, 19anim12i 616 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦)))
21 3anass 1093 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) ↔ (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦)))
2220, 21sylibr 237 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
23 uspgredg2vtxeu 27094 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
24 reueq1 3323 . . . . . . . 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 237 . . . . . 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 2834 . . . . . . . . . . 11 (𝑒 = 𝑥 → (𝑁𝑒𝑁𝑥))
2928, 3elrab2 3603 . . . . . . . . . 10 (𝑥𝐴 ↔ (𝑥𝐸𝑁𝑥))
302eleq2i 2842 . . . . . . . . . . . 12 (𝑥𝐸𝑥 ∈ (Edg‘𝐺))
3130biimpi 219 . . . . . . . . . . 11 (𝑥𝐸𝑥 ∈ (Edg‘𝐺))
3231anim1i 618 . . . . . . . . . 10 ((𝑥𝐸𝑁𝑥) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3329, 32sylbi 220 . . . . . . . . 9 (𝑥𝐴 → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3433adantl 486 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3512, 34anim12i 616 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥)))
36 3anass 1093 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥) ↔ (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥)))
3735, 36sylibr 237 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
38 uspgredg2vtxeu 27094 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
39 reueq1 3323 . . . . . . . 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 237 . . . . . 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 7127 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) ∧ (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧})) → 𝑦 = 𝑥)
4443ex 417 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
4544ralrimivva 3118 . 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 2763 . . . 4 (𝑦 = 𝑥 → (𝑦 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑧}))
4847riotabidv 7103 . . 3 (𝑦 = 𝑥 → (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}))
4946, 48f1mpt 7004 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉 ∧ ∀𝑦𝐴𝑥𝐴 ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥)))
506, 45, 49sylanbrc 587 1 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3068  ∃!wreu 3070  {crab 3072  {cpr 4517   ↦ cmpt 5105  –1-1→wf1 6325  ‘cfv 6328  ℩crio 7100  Vtxcvtx 26873  Edgcedg 26924  USPGraphcuspgr 27025 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rmo 3076  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-1st 7686  df-2nd 7687  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-en 8521  df-dom 8522  df-sdom 8523  df-fin 8524  df-dju 9348  df-card 9386  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-nn 11660  df-2 11722  df-n0 11920  df-xnn0 11992  df-z 12006  df-uz 12268  df-fz 12925  df-hash 13726  df-edg 26925  df-upgr 26959  df-uspgr 27027 This theorem is referenced by:  uspgredgleord  27106
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