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Theorem uslgraf1oedg 25888
 Description: The edge function of an undirected simple graph with loops is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.)
Assertion
Ref Expression
uslgraf1oedg (𝑉 USLGrph 𝐸𝐸:dom 𝐸1-1-onto→(𝑉Edges𝐸))

Proof of Theorem uslgraf1oedg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 uslgraf 25874 . 2 (𝑉 USLGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
2 f1f1orn 6061 . . 3 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
3 edguslgra 25871 . . . . 5 (𝑉 USLGrph 𝐸 → (𝑉Edges𝐸) = ran 𝐸)
43eqcomd 2616 . . . 4 (𝑉 USLGrph 𝐸 → ran 𝐸 = (𝑉Edges𝐸))
5 f1oeq3 6042 . . . 4 (ran 𝐸 = (𝑉Edges𝐸) → (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1-onto→(𝑉Edges𝐸)))
64, 5syl 17 . . 3 (𝑉 USLGrph 𝐸 → (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1-onto→(𝑉Edges𝐸)))
72, 6syl5ibcom 234 . 2 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → (𝑉 USLGrph 𝐸𝐸:dom 𝐸1-1-onto→(𝑉Edges𝐸)))
81, 7mpcom 37 1 (𝑉 USLGrph 𝐸𝐸:dom 𝐸1-1-onto→(𝑉Edges𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  {crab 2900   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ran crn 5039  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ≤ cle 9954  2c2 10947  #chash 12979   USLGrph cuslg 25858  Edgescedg 25860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-2nd 7060  df-uslgra 25861  df-edg 25865 This theorem is referenced by: (None)
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