Theorem opiedgval 27376

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) = (2nd ‘𝐺))

Proof of Theorem opiedgval

Step	Hyp	Ref	Expression
1		iedgval 27371	. 2 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))
2		iftrue 4465	. 2 ⊢ (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (2nd ‘𝐺))
3	1, 2	eqtrid 2790	1 ⊢ (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd ‘𝐺))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ifcif 4459   × cxp 5587  ‘cfv 6433  2^nd c2nd 7830  .efcedgf 27356  iEdgciedg 27367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-iedg 27369

This theorem is referenced by:  opiedgfv  27377