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Theorem uspgredg2v 28214
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 28213 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑦𝐴) → (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
54ralrimiva 3140 . . 3 (𝐺 ∈ USPGraph → ∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
65adantr 482 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → ∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
7 preq2 4696 . . . . . . 7 (𝑧 = 𝑛 → {𝑁, 𝑧} = {𝑁, 𝑛})
87eqeq2d 2744 . . . . . 6 (𝑧 = 𝑛 → (𝑦 = {𝑁, 𝑧} ↔ 𝑦 = {𝑁, 𝑛}))
98cbvriotavw 7324 . . . . 5 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑛𝑉 𝑦 = {𝑁, 𝑛})
107eqeq2d 2744 . . . . . 6 (𝑧 = 𝑛 → (𝑥 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑛}))
1110cbvriotavw 7324 . . . . 5 (𝑧𝑉 𝑥 = {𝑁, 𝑧}) = (𝑛𝑉 𝑥 = {𝑁, 𝑛})
12 simpl 484 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐺 ∈ USPGraph)
13 eleq2w 2818 . . . . . . . . . . 11 (𝑒 = 𝑦 → (𝑁𝑒𝑁𝑦))
1413, 3elrab2 3649 . . . . . . . . . 10 (𝑦𝐴 ↔ (𝑦𝐸𝑁𝑦))
152eleq2i 2826 . . . . . . . . . . . 12 (𝑦𝐸𝑦 ∈ (Edg‘𝐺))
1615biimpi 215 . . . . . . . . . . 11 (𝑦𝐸𝑦 ∈ (Edg‘𝐺))
1716anim1i 616 . . . . . . . . . 10 ((𝑦𝐸𝑁𝑦) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
1814, 17sylbi 216 . . . . . . . . 9 (𝑦𝐴 → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
1918adantr 482 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
2012, 19anim12i 614 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦)))
21 3anass 1096 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) ↔ (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦)))
2220, 21sylibr 233 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
23 uspgredg2vtxeu 28210 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
24 reueq1 3391 . . . . . . . 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 233 . . . . . 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 2818 . . . . . . . . . . 11 (𝑒 = 𝑥 → (𝑁𝑒𝑁𝑥))
2928, 3elrab2 3649 . . . . . . . . . 10 (𝑥𝐴 ↔ (𝑥𝐸𝑁𝑥))
302eleq2i 2826 . . . . . . . . . . . 12 (𝑥𝐸𝑥 ∈ (Edg‘𝐺))
3130biimpi 215 . . . . . . . . . . 11 (𝑥𝐸𝑥 ∈ (Edg‘𝐺))
3231anim1i 616 . . . . . . . . . 10 ((𝑥𝐸𝑁𝑥) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3329, 32sylbi 216 . . . . . . . . 9 (𝑥𝐴 → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3433adantl 483 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3512, 34anim12i 614 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥)))
36 3anass 1096 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥) ↔ (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥)))
3735, 36sylibr 233 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
38 uspgredg2vtxeu 28210 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
39 reueq1 3391 . . . . . . . 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 233 . . . . . 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 7341 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) ∧ (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧})) → 𝑦 = 𝑥)
4443ex 414 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
4544ralrimivva 3194 . 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 2737 . . . 4 (𝑦 = 𝑥 → (𝑦 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑧}))
4847riotabidv 7316 . . 3 (𝑦 = 𝑥 → (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}))
4946, 48f1mpt 7209 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉 ∧ ∀𝑦𝐴𝑥𝐴 ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥)))
506, 45, 49sylanbrc 584 1 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  ∃!wreu 3350  {crab 3406  {cpr 4589  cmpt 5189  1-1wf1 6494  cfv 6497  crio 7313  Vtxcvtx 27989  Edgcedg 28040  USPGraphcuspgr 28141
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-oadd 8417  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-dju 9842  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-n0 12419  df-xnn0 12491  df-z 12505  df-uz 12769  df-fz 13431  df-hash 14237  df-edg 28041  df-upgr 28075  df-uspgr 28143
This theorem is referenced by:  uspgredgleord  28222
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