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Theorem usgrf1oedg 26269
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 2748 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgrf1oedg.i . . . 4 𝐼 = (iEdg‘𝐺)
31, 2usgrf 26220 . . 3 (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4 f1f1orn 6297 . . 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 26111 . . . . . 6 (Edg‘𝐺) = ran (iEdg‘𝐺)
87a1i 11 . . . . 5 (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
92eqcomi 2757 . . . . . 6 (iEdg‘𝐺) = 𝐼
109rneqi 5495 . . . . 5 ran (iEdg‘𝐺) = ran 𝐼
118, 10syl6eq 2798 . . . 4 (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼)
126, 11syl5eq 2794 . . 3 (𝐺 ∈ USGraph → 𝐸 = ran 𝐼)
1312f1oeq3d 6283 . 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 1620  wcel 2127  {crab 3042  cdif 3700  c0 4046  𝒫 cpw 4290  {csn 4309  dom cdm 5254  ran crn 5255  1-1wf1 6034  1-1-ontowf1o 6036  cfv 6037  2c2 11233  chash 13282  Vtxcvtx 26044  iEdgciedg 26045  Edgcedg 26109  USGraphcusgr 26214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-edg 26110  df-usgr 26216
This theorem is referenced by:  usgr2trlncl  26837
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