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Theorem opiedgval 26898
 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 26893 . 2 (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺))
2 iftrue 4426 . 2 (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)) = (2nd𝐺))
31, 2syl5eq 2805 1 (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ifcif 4420   × cxp 5522  ‘cfv 6335  2nd c2nd 7692  .efcedgf 26881  iEdgciedg 26889 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-iedg 26891 This theorem is referenced by:  opiedgfv  26899
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