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Theorem uspgrf1oedg 29205
Description: The edge function of a simple pseudograph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
Hypothesis
Ref Expression
usgrf1o.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
uspgrf1oedg (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))

Proof of Theorem uspgrf1oedg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgrf1o.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2uspgrf 29186 . 2 (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4 f1f1orn 6860 . . 3 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
52rneqi 5951 . . . . 5 ran 𝐸 = ran (iEdg‘𝐺)
6 edgval 29081 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
75, 6eqtr4i 2766 . . . 4 ran 𝐸 = (Edg‘𝐺)
8 f1oeq3 6839 . . . 4 (ran 𝐸 = (Edg‘𝐺) → (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1-onto→(Edg‘𝐺)))
97, 8ax-mp 5 . . 3 (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))
104, 9sylib 218 . 2 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))
113, 10syl 17 1 (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1-onto→(Edg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  {crab 3433  cdif 3960  c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  dom cdm 5689  ran crn 5690  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  cle 11294  2c2 12319  chash 14366  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079  USPGraphcuspgr 29180
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-uspgr 29182
This theorem is referenced by:  uspgredgiedg  29207  uspgriedgedg  29208  uspgr2wlkeq  29679  wlkiswwlks2lem4  29902  wlkiswwlks2lem5  29903  clwlkclwwlk  30031  isuspgrim0lem  47809  uspgrlimlem1  47891  uspgrlimlem2  47892  uspgrlimlem3  47893  uspgrlimlem4  47894  uspgrlim  47895
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