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Theorem usgredgedga 40457
 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 by AV, 18-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.)
Hypotheses
Ref Expression
ushgredgedga.e 𝐸 = (Edg‘𝐺)
ushgredgedga.i 𝐼 = (iEdg‘𝐺)
ushgredgedga.v 𝑉 = (Vtx‘𝐺)
ushgredgedga.a 𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}
ushgredgedga.b 𝐵 = {𝑒𝐸𝑁𝑒}
ushgredgedga.f 𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))
Assertion
Ref Expression
usgredgedga ((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 usgredgedga
StepHypRef Expression
1 usgruspgr 40408 . . 3 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
2 uspgrushgr 40405 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph )
31, 2syl 17 . 2 (𝐺 ∈ USGraph → 𝐺 ∈ USHGraph )
4 ushgredgedga.e . . 3 𝐸 = (Edg‘𝐺)
5 ushgredgedga.i . . 3 𝐼 = (iEdg‘𝐺)
6 ushgredgedga.v . . 3 𝑉 = (Vtx‘𝐺)
7 ushgredgedga.a . . 3 𝐴 = {𝑖 ∈ dom 𝐼𝑁 ∈ (𝐼𝑖)}
8 ushgredgedga.b . . 3 𝐵 = {𝑒𝐸𝑁𝑒}
9 ushgredgedga.f . . 3 𝐹 = (𝑥𝐴 ↦ (𝐼𝑥))
104, 5, 6, 7, 8, 9ushgredgedga 40456 . 2 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
113, 10sylan 487 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ↦ cmpt 4643  dom cdm 5038  –1-1-onto→wf1o 5803  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   USHGraph cushgr 25723  Edgcedga 25792   USPGraph cuspgr 40378   USGraph cusgr 40379 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890 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-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-2 10956  df-uhgr 25724  df-ushgr 25725  df-edga 25793  df-uspgr 40380  df-usgr 40381 This theorem is referenced by:  usgredgaleordALT  40461
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