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Theorem usgrf1oedg 29239
Description: The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed 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 2735 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgrf1oedg.i . . . 4 𝐼 = (iEdg‘𝐺)
31, 2usgrf 29187 . . 3 (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4 f1f1orn 6860 . . 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 29081 . . . . . 6 (Edg‘𝐺) = ran (iEdg‘𝐺)
87a1i 11 . . . . 5 (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
92eqcomi 2744 . . . . . 6 (iEdg‘𝐺) = 𝐼
109rneqi 5951 . . . . 5 ran (iEdg‘𝐺) = ran 𝐼
118, 10eqtrdi 2791 . . . 4 (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼)
126, 11eqtrid 2787 . . 3 (𝐺 ∈ USGraph → 𝐸 = ran 𝐼)
1312f1oeq3d 6846 . 2 (𝐺 ∈ USGraph → (𝐼:dom 𝐼1-1-onto𝐸𝐼:dom 𝐼1-1-onto→ran 𝐼))
145, 13mpbird 257 1 (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1-onto𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  cdif 3960  c0 4339  𝒫 cpw 4605  {csn 4631  dom cdm 5689  ran crn 5690  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  2c2 12319  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079  USGraphcusgr 29181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-edg 29080  df-usgr 29183
This theorem is referenced by:  usgr2trlncl  29793
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