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Theorem usgredgedg 29087
Description: In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
Hypotheses
Ref Expression
ushgredgedg.e 𝐸 = (Edg‘𝐺)
ushgredgedg.i 𝐼 = (iEdg‘𝐺)
ushgredgedg.v 𝑉 = (Vtx‘𝐺)
ushgredgedg.a 𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}
ushgredgedg.b 𝐵 = {𝑒𝐸𝑁𝑒}
ushgredgedg.f 𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))
Assertion
Ref Expression
usgredgedg ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
Distinct variable groups:   𝐵,𝑒   𝑒,𝐸,𝑖   𝑒,𝐺,𝑖,𝑥   𝑒,𝐼,𝑖,𝑥   𝑒,𝑁,𝑖,𝑥   𝑒,𝑉,𝑖,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑒,𝑖)   𝐵(𝑥,𝑖)   𝐸(𝑥)   𝐹(𝑥,𝑒,𝑖)

Proof of Theorem usgredgedg
StepHypRef Expression
1 usgruspgr 29037 . . 3 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2 uspgrushgr 29034 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
31, 2syl 17 . 2 (𝐺 ∈ USGraph → 𝐺 ∈ USHGraph)
4 ushgredgedg.e . . 3 𝐸 = (Edg‘𝐺)
5 ushgredgedg.i . . 3 𝐼 = (iEdg‘𝐺)
6 ushgredgedg.v . . 3 𝑉 = (Vtx‘𝐺)
7 ushgredgedg.a . . 3 𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}
8 ushgredgedg.b . . 3 𝐵 = {𝑒𝐸𝑁𝑒}
9 ushgredgedg.f . . 3 𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))
104, 5, 6, 7, 8, 9ushgredgedg 29086 . 2 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
113, 10sylan 578 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {crab 3419  cmpt 5226  dom cdm 5672  1-1-ontowf1o 6542  cfv 6543  Vtxcvtx 28853  iEdgciedg 28854  Edgcedg 28904  USHGraphcushgr 28914  USPGraphcuspgr 29005  USGraphcusgr 29006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-i2m1 11206  ax-1ne0 11207  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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-ov 7419  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-2 12305  df-edg 28905  df-uhgr 28915  df-ushgr 28916  df-uspgr 29007  df-usgr 29008
This theorem is referenced by:  usgredgleordALT  29091
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