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Theorem usgredgedg 28995
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 28946 . . 3 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2 uspgrushgr 28943 . . 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 28994 . 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 1533  wcel 2098  {crab 3426  cmpt 5224  dom cdm 5669  1-1-ontowf1o 6536  cfv 6537  Vtxcvtx 28764  iEdgciedg 28765  Edgcedg 28815  USHGraphcushgr 28825  USPGraphcuspgr 28916  USGraphcusgr 28917
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-i2m1 11180  ax-1ne0 11181  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-2 12279  df-edg 28816  df-uhgr 28826  df-ushgr 28827  df-uspgr 28918  df-usgr 28919
This theorem is referenced by:  usgredgleordALT  28999
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