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Theorem uspgredg2v 40451
 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 40450 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑦𝐴) → (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
54ralrimiva 2949 . . 3 (𝐺 ∈ USPGraph → ∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
65adantr 480 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → ∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉)
7 preq2 4213 . . . . . . 7 (𝑧 = 𝑛 → {𝑁, 𝑧} = {𝑁, 𝑛})
87eqeq2d 2620 . . . . . 6 (𝑧 = 𝑛 → (𝑦 = {𝑁, 𝑧} ↔ 𝑦 = {𝑁, 𝑛}))
98cbvriotav 6522 . . . . 5 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑛𝑉 𝑦 = {𝑁, 𝑛})
107eqeq2d 2620 . . . . . 6 (𝑧 = 𝑛 → (𝑥 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑛}))
1110cbvriotav 6522 . . . . 5 (𝑧𝑉 𝑥 = {𝑁, 𝑧}) = (𝑛𝑉 𝑥 = {𝑁, 𝑛})
12 simpl 472 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐺 ∈ USPGraph )
13 eleq2 2677 . . . . . . . . . . 11 (𝑒 = 𝑦 → (𝑁𝑒𝑁𝑦))
1413, 3elrab2 3333 . . . . . . . . . 10 (𝑦𝐴 ↔ (𝑦𝐸𝑁𝑦))
152eleq2i 2680 . . . . . . . . . . . 12 (𝑦𝐸𝑦 ∈ (Edg‘𝐺))
1615biimpi 205 . . . . . . . . . . 11 (𝑦𝐸𝑦 ∈ (Edg‘𝐺))
1716anim1i 590 . . . . . . . . . 10 ((𝑦𝐸𝑁𝑦) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
1814, 17sylbi 206 . . . . . . . . 9 (𝑦𝐴 → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
1918adantr 480 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
2012, 19anim12i 588 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦)))
21 3anass 1035 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) ↔ (𝐺 ∈ USPGraph ∧ (𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦)))
2220, 21sylibr 223 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦))
23 uspgredg2vtxeu 40447 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑦 = {𝑁, 𝑛})
24 reueq1 3117 . . . . . . . 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 223 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑦 ∈ (Edg‘𝐺) ∧ 𝑁𝑦) → ∃!𝑛𝑉 𝑦 = {𝑁, 𝑛})
2722, 26syl 17 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → ∃!𝑛𝑉 𝑦 = {𝑁, 𝑛})
28 eleq2 2677 . . . . . . . . . . 11 (𝑒 = 𝑥 → (𝑁𝑒𝑁𝑥))
2928, 3elrab2 3333 . . . . . . . . . 10 (𝑥𝐴 ↔ (𝑥𝐸𝑁𝑥))
302eleq2i 2680 . . . . . . . . . . . 12 (𝑥𝐸𝑥 ∈ (Edg‘𝐺))
3130biimpi 205 . . . . . . . . . . 11 (𝑥𝐸𝑥 ∈ (Edg‘𝐺))
3231anim1i 590 . . . . . . . . . 10 ((𝑥𝐸𝑁𝑥) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3329, 32sylbi 206 . . . . . . . . 9 (𝑥𝐴 → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3433adantl 481 . . . . . . . 8 ((𝑦𝐴𝑥𝐴) → (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
3512, 34anim12i 588 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥)))
36 3anass 1035 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥) ↔ (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥)))
3735, 36sylibr 223 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → (𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥))
38 uspgredg2vtxeu 40447 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑥 ∈ (Edg‘𝐺) ∧ 𝑁𝑥) → ∃!𝑛 ∈ (Vtx‘𝐺)𝑥 = {𝑁, 𝑛})
39 reueq1 3117 . . . . . . . 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 223 . . . . . 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 40338 . . . 4 ((((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) ∧ (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧})) → 𝑦 = 𝑥)
4443ex 449 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁𝑉) ∧ (𝑦𝐴𝑥𝐴)) → ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥))
4544ralrimivva 2954 . 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 2614 . . . 4 (𝑦 = 𝑥 → (𝑦 = {𝑁, 𝑧} ↔ 𝑥 = {𝑁, 𝑧}))
4847riotabidv 6513 . . 3 (𝑦 = 𝑥 → (𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}))
4946, 48f1mpt 6419 . 2 (𝐹:𝐴1-1𝑉 ↔ (∀𝑦𝐴 (𝑧𝑉 𝑦 = {𝑁, 𝑧}) ∈ 𝑉 ∧ ∀𝑦𝐴𝑥𝐴 ((𝑧𝑉 𝑦 = {𝑁, 𝑧}) = (𝑧𝑉 𝑥 = {𝑁, 𝑧}) → 𝑦 = 𝑥)))
506, 45, 49sylanbrc 695 1 ((𝐺 ∈ USPGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1𝑉)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃!wreu 2898  {crab 2900  {cpr 4127   ↦ cmpt 4643  –1-1→wf1 5801  ‘cfv 5804  ℩crio 6510  Vtxcvtx 25673  Edgcedga 25792   USPGraph cuspgr 40378 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-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-rmo 2904  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-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  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-hash 12980  df-upgr 25749  df-edga 25793  df-uspgr 40380 This theorem is referenced by:  uspgredgaleord  40459
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