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Theorem opiedgval 28530
Description: The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020.)
Assertion
Ref Expression
opiedgval (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd𝐺))

Proof of Theorem opiedgval
StepHypRef Expression
1 iedgval 28525 . 2 (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺))
2 iftrue 4535 . 2 (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)) = (2nd𝐺))
31, 2eqtrid 2783 1 (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  Vcvv 3473  ifcif 4529   × cxp 5675  cfv 6544  2nd c2nd 7977  .efcedgf 28510  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
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-iota 6496  df-fun 6546  df-fv 6552  df-iedg 28523
This theorem is referenced by:  opiedgfv  28531
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