Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgredgedg Structured version   Visualization version   GIF version

Theorem usgredgedg 26321
 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
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 26272 . . 3 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2 uspgrushgr 26269 . . 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 26320 . 2 ((𝐺 ∈ USHGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
113, 10sylan 489 1 ((𝐺 ∈ USGraph ∧ 𝑁𝑉) → 𝐹:𝐴1-1-onto𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {crab 3054   ↦ cmpt 4881  dom cdm 5266  –1-1-onto→wf1o 6048  ‘cfv 6049  Vtxcvtx 26073  iEdgciedg 26074  Edgcedg 26138  USHGraphcushgr 26151  USPGraphcuspgr 26242  USGraphcusgr 26243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-i2m1 10196  ax-1ne0 10197  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-2 11271  df-edg 26139  df-uhgr 26152  df-ushgr 26153  df-uspgr 26244  df-usgr 26245 This theorem is referenced by:  usgredgleordALT  26325
 Copyright terms: Public domain W3C validator