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Theorem uspgrf1oedg 28867
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 2731 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgrf1o.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2uspgrf 28848 . 2 (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4 f1f1orn 6844 . . 3 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
52rneqi 5936 . . . . 5 ran 𝐸 = ran (iEdg‘𝐺)
6 edgval 28743 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
75, 6eqtr4i 2762 . . . 4 ran 𝐸 = (Edg‘𝐺)
8 f1oeq3 6823 . . . 4 (ran 𝐸 = (Edg‘𝐺) → (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1-onto→(Edg‘𝐺)))
97, 8ax-mp 5 . . 3 (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))
104, 9sylib 217 . 2 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))
113, 10syl 17 1 (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  {crab 3431  cdif 3945  c0 4322  𝒫 cpw 4602  {csn 4628   class class class wbr 5148  dom cdm 5676  ran crn 5677  1-1wf1 6540  1-1-ontowf1o 6542  cfv 6543  cle 11256  2c2 12274  chash 14297  Vtxcvtx 28690  iEdgciedg 28691  Edgcedg 28741  USPGraphcuspgr 28842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-edg 28742  df-uspgr 28844
This theorem is referenced by:  uspgr2wlkeq  29337  wlkiswwlks2lem4  29560  wlkiswwlks2lem5  29561  clwlkclwwlk  29689
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