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Theorem usgrf1oedg 26144
 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 2651 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgrf1oedg.i . . . 4 𝐼 = (iEdg‘𝐺)
31, 2usgrf 26095 . . 3 (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) = 2})
4 f1f1orn 6186 . . 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 25986 . . . . . 6 (Edg‘𝐺) = ran (iEdg‘𝐺)
87a1i 11 . . . . 5 (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
92eqcomi 2660 . . . . . 6 (iEdg‘𝐺) = 𝐼
109rneqi 5384 . . . . 5 ran (iEdg‘𝐺) = ran 𝐼
118, 10syl6eq 2701 . . . 4 (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼)
126, 11syl5eq 2697 . . 3 (𝐺 ∈ USGraph → 𝐸 = ran 𝐼)
1312f1oeq3d 6172 . 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   = wceq 1523   ∈ wcel 2030  {crab 2945   ∖ cdif 3604  ∅c0 3948  𝒫 cpw 4191  {csn 4210  dom cdm 5143  ran crn 5144  –1-1→wf1 5923  –1-1-onto→wf1o 5925  ‘cfv 5926  2c2 11108  #chash 13157  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  USGraphcusgr 26089 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-edg 25985  df-usgr 26091 This theorem is referenced by:  usgr2trlncl  26712
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