Theorem opiedgval 28933

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 28928	. 2 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))
2		iftrue 4494	. 2 ⊢ (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (2nd ‘𝐺))
3	1, 2	eqtrid 2776	1 ⊢ (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd ‘𝐺))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ifcif 4488   × cxp 5636  ‘cfv 6511  2^nd c2nd 7967  .efcedgf 28915  iEdgciedg 28924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-iedg 28926

This theorem is referenced by:  opiedgfv  28934