Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uspgrf1oedg Structured version   Visualization version   GIF version

Theorem uspgrf1oedg 40403
 Description: The edge function of a simple pseudograph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
Hypothesis
Ref Expression
usgrf1o.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
uspgrf1oedg (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))

Proof of Theorem uspgrf1oedg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgrf1o.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2uspgrf 40384 . 2 (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
4 f1f1orn 6061 . . 3 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
5 edgaval 25794 . . . . 5 (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
62rneqi 5273 . . . . 5 ran 𝐸 = ran (iEdg‘𝐺)
75, 6syl6reqr 2663 . . . 4 (𝐺 ∈ USPGraph → ran 𝐸 = (Edg‘𝐺))
8 f1oeq3 6042 . . . 4 (ran 𝐸 = (Edg‘𝐺) → (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1-onto→(Edg‘𝐺)))
97, 8syl 17 . . 3 (𝐺 ∈ USPGraph → (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1-onto→(Edg‘𝐺)))
104, 9syl5ibcom 234 . 2 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺)))
113, 10mpcom 37 1 (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ran crn 5039  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792   USPGraph cuspgr 40378 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-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-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-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  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-edga 25793  df-uspgr 40380 This theorem is referenced by:  uspgr2wlkeq  40854  1wlkiswwlks2lem4  41069  1wlkiswwlks2lem5  41070  clwlkclwwlk  41211
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