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Theorem uspgrf1oedg 29191
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 2736 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 usgrf1o.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2uspgrf 29172 . 2 (𝐺 ∈ USPGraph → 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4 f1f1orn 6858 . . 3 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
52rneqi 5947 . . . . 5 ran 𝐸 = ran (iEdg‘𝐺)
6 edgval 29067 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
75, 6eqtr4i 2767 . . . 4 ran 𝐸 = (Edg‘𝐺)
8 f1oeq3 6837 . . . 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 1539  wcel 2107  {crab 3435  cdif 3947  c0 4332  𝒫 cpw 4599  {csn 4625   class class class wbr 5142  dom cdm 5684  ran crn 5685  1-1wf1 6557  1-1-ontowf1o 6559  cfv 6560  cle 11297  2c2 12322  chash 14370  Vtxcvtx 29014  iEdgciedg 29015  Edgcedg 29065  USPGraphcuspgr 29166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-edg 29066  df-uspgr 29168
This theorem is referenced by:  uspgredgiedg  29193  uspgriedgedg  29194  uspgr2wlkeq  29665  wlkiswwlks2lem4  29893  wlkiswwlks2lem5  29894  clwlkclwwlk  30022  isuspgrim0lem  47876  uspgrlimlem1  47960  uspgrlimlem2  47961  uspgrlimlem3  47962  uspgrlimlem4  47963  uspgrlim  47964
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