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Theorem opiedgfv 28531
Description: The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
Assertion
Ref Expression
opiedgfv ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Proof of Theorem opiedgfv
StepHypRef Expression
1 opelvvg 5718 . . 3 ((𝑉𝑋𝐸𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2 opiedgval 28530 . . 3 (⟨𝑉, 𝐸⟩ ∈ (V × V) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
31, 2syl 17 . 2 ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = (2nd ‘⟨𝑉, 𝐸⟩))
4 op2ndg 7991 . 2 ((𝑉𝑋𝐸𝑌) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
53, 4eqtrd 2771 1 ((𝑉𝑋𝐸𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  cop 4635   × cxp 5675  cfv 6544  2nd c2nd 7977  iEdgciedg 28521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-2nd 7979  df-iedg 28523
This theorem is referenced by:  opiedgov  28532  opiedgfvi  28534  gropd  28555  edgopval  28575  isuhgrop  28594  uhgrunop  28599  upgrop  28618  upgr0eop  28638  upgr1eop  28639  upgrunop  28643  umgrunop  28645  isuspgrop  28685  isusgrop  28686  ausgrusgrb  28689  usgr0eop  28767  uspgr1eop  28768  usgr1eop  28771  usgrexmpllem  28781  uhgrspan1lem3  28823  upgrres1lem3  28833  fusgrfisbase  28849  fusgrfisstep  28850  usgrexi  28962  cusgrexi  28964  p1evtxdeqlem  29033  p1evtxdeq  29034  p1evtxdp1  29035  uspgrloopiedg  29038  umgr2v2eiedg  29044  wlk2v2e  29674  eupthvdres  29752  eupth2lemb  29754  konigsbergiedg  29764  strisomgrop  46808  ushrisomgr  46809
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