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Theorem usgrf1oedg 26916
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 2818 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgrf1oedg.i . . . 4 𝐼 = (iEdg‘𝐺)
31, 2usgrf 26867 . . 3 (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4 f1f1orn 6619 . . 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 26761 . . . . . 6 (Edg‘𝐺) = ran (iEdg‘𝐺)
87a1i 11 . . . . 5 (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
92eqcomi 2827 . . . . . 6 (iEdg‘𝐺) = 𝐼
109rneqi 5800 . . . . 5 ran (iEdg‘𝐺) = ran 𝐼
118, 10syl6eq 2869 . . . 4 (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼)
126, 11syl5eq 2865 . . 3 (𝐺 ∈ USGraph → 𝐸 = ran 𝐼)
1312f1oeq3d 6605 . 2 (𝐺 ∈ USGraph → (𝐼:dom 𝐼1-1-onto𝐸𝐼:dom 𝐼1-1-onto→ran 𝐼))
145, 13mpbird 258 1 (𝐺 ∈ USGraph → 𝐼:dom 𝐼1-1-onto𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {crab 3139  cdif 3930  c0 4288  𝒫 cpw 4535  {csn 4557  dom cdm 5548  ran crn 5549  1-1wf1 6345  1-1-ontowf1o 6347  cfv 6348  2c2 11680  chash 13678  Vtxcvtx 26708  iEdgciedg 26709  Edgcedg 26759  USGraphcusgr 26861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-edg 26760  df-usgr 26863
This theorem is referenced by:  usgr2trlncl  27468
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