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Theorem usgrf1oedg 25992
Description: The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by by AV, 18-Oct-2020.)
Hypotheses
Ref Expression
usgrf1oedg.i 𝐼 = (iEdg‘𝐺)
usgrf1oedg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
usgrf1oedg (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1-onto𝐸)

Proof of Theorem usgrf1oedg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgrf1oedg.i . . . 4 𝐼 = (iEdg‘𝐺)
31, 2usgrf 25943 . . 3 (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})
4 f1f1orn 6105 . . 3 (𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐼:dom 𝐼1-1-onto→ran 𝐼)
53, 4syl 17 . 2 (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1-onto→ran 𝐼)
6 usgrf1oedg.e . . . 4 𝐸 = (Edg‘𝐺)
7 edgval 25841 . . . . 5 (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
82eqcomi 2630 . . . . . 6 (iEdg‘𝐺) = 𝐼
98rneqi 5312 . . . . 5 ran (iEdg‘𝐺) = ran 𝐼
107, 9syl6eq 2671 . . . 4 (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼)
116, 10syl5eq 2667 . . 3 (𝐺 ∈ USGraph → 𝐸 = ran 𝐼)
12 f1oeq3 6086 . . 3 (𝐸 = ran 𝐼 → (𝐼:dom 𝐼1-1-onto𝐸𝐼:dom 𝐼1-1-onto→ran 𝐼))
1311, 12syl 17 . 2 (𝐺 ∈ USGraph → (𝐼:dom 𝐼1-1-onto𝐸𝐼:dom 𝐼1-1-onto→ran 𝐼))
145, 13mpbird 247 1 (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1-onto𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  {crab 2911  cdif 3552  c0 3891  𝒫 cpw 4130  {csn 4148  dom cdm 5074  ran crn 5075  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  2c2 11014  #chash 13057  Vtxcvtx 25774  iEdgciedg 25775  Edgcedg 25839   USGraph cusgr 25937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-edg 25840  df-usgr 25939
This theorem is referenced by:  usgr2trlncl  26525
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