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Theorem usgredgedg 29056
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 29006 . . 3 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2 uspgrushgr 29003 . . 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 29055 . 2 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
113, 10sylan 579 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  {crab 3429  cmpt 5231  dom cdm 5678  1-1-ontowf1o 6547  cfv 6548  Vtxcvtx 28822  iEdgciedg 28823  Edgcedg 28873  USHGraphcushgr 28883  USPGraphcuspgr 28974  USGraphcusgr 28975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-i2m1 11207  ax-1ne0 11208  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-2 12306  df-edg 28874  df-uhgr 28884  df-ushgr 28885  df-uspgr 28976  df-usgr 28977
This theorem is referenced by:  usgredgleordALT  29060
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