Theorem opiedgval 29041

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 29036	. 2 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))
2		iftrue 4554	. 2 ⊢ (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (2nd ‘𝐺))
3	1, 2	eqtrid 2792	1 ⊢ (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd ‘𝐺))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ifcif 4548   × cxp 5698  ‘cfv 6573  2^nd c2nd 8029  .efcedgf 29021  iEdgciedg 29032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-iedg 29034

This theorem is referenced by:  opiedgfv  29042