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Theorem uspgredg2v 28914
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 28913 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑦𝐴) → (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
54ralrimiva 3145 . . 3 (𝐺 ∈ USPGraph → ∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
65adantr 480 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → ∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
7 preq2 4738 . . . . . . 7 (𝑧 = 𝑛 → {𝑁, 𝑧} = {𝑁, 𝑛})
87eqeq2d 2742 . . . . . 6 (𝑧 = 𝑛 → (𝑦 = {𝑁, 𝑧} ↔ 𝑦 = {𝑁, 𝑛}))
98cbvriotavw 7378 . . . . 5 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑛𝑉 𝑦 = {𝑁, 𝑛})
107eqeq2d 2742 . . . . . 6 (𝑧 = 𝑛 → (𝑥 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑛}))
1110cbvriotavw 7378 . . . . 5 (𝑧𝑉 𝑥 = {𝑁, 𝑧}) = (𝑛𝑉 𝑥 = {𝑁, 𝑛})
12 simpl 482 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐺 ∈ USPGraph)
13 eleq2w 2816 . . . . . . . . . . 11 (𝑒 = 𝑦 → (𝑁𝑒𝑁𝑦))
1413, 3elrab2 3686 . . . . . . . . . 10 (𝑦𝐴 ↔ (𝑦𝐸𝑁𝑦))
152eleq2i 2824 . . . . . . . . . . . 12 (𝑦𝐸𝑦 ∈ (Edg‘𝐺))
1615biimpi 215 . . . . . . . . . . 11 (𝑦𝐸𝑦 ∈ (Edg‘𝐺))
1716anim1i 614 . . . . . . . . . 10 ((𝑦𝐸𝑁𝑦) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
1814, 17sylbi 216 . . . . . . . . 9 (𝑦𝐴 → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
1918adantr 480 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
2012, 19anim12i 612 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦)))
21 3anass 1094 . . . . . . 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 28910 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
24 reueq1 3414 . . . . . . . 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 2816 . . . . . . . . . . 11 (𝑒 = 𝑥 → (𝑁𝑒𝑁𝑥))
2928, 3elrab2 3686 . . . . . . . . . 10 (𝑥𝐴 ↔ (𝑥𝐸𝑁𝑥))
302eleq2i 2824 . . . . . . . . . . . 12 (𝑥𝐸𝑥 ∈ (Edg‘𝐺))
3130biimpi 215 . . . . . . . . . . 11 (𝑥𝐸𝑥 ∈ (Edg‘𝐺))
3231anim1i 614 . . . . . . . . . 10 ((𝑥𝐸𝑁𝑥) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3329, 32sylbi 216 . . . . . . . . 9 (𝑥𝐴 → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3433adantl 481 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3512, 34anim12i 612 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥)))
36 3anass 1094 . . . . . . 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 28910 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
39 reueq1 3414 . . . . . . . 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 7395 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) ∧ (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧})) → 𝑦 = 𝑥)
4443ex 412 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
4544ralrimivva 3199 . 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 2735 . . . 4 (𝑦 = 𝑥 → (𝑦 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑧}))
4847riotabidv 7370 . . 3 (𝑦 = 𝑥 → (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}))
4946, 48f1mpt 7263 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉 ∧ ∀𝑦𝐴𝑥𝐴 ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥)))
506, 45, 49sylanbrc 582 1 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  ∃!wreu 3373  {crab 3431  {cpr 4630  cmpt 5231  1-1wf1 6540  cfv 6543  crio 7367  Vtxcvtx 28689  Edgcedg 28740  USPGraphcuspgr 28841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-oadd 8476  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-dju 9902  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-n0 12480  df-xnn0 12552  df-z 12566  df-uz 12830  df-fz 13492  df-hash 14298  df-edg 28741  df-upgr 28775  df-uspgr 28843
This theorem is referenced by:  uspgredgleord  28922
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