Theorem opiedgval 26107

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 26100	. 2 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))
2		iftrue 4232	. 2 ⊢ (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (2nd ‘𝐺))
3	1, 2	syl5eq 2817	1 ⊢ (𝐺 ∈ (V × V) → (iEdg‘𝐺) = (2nd ‘𝐺))

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  Vcvv 3351  ifcif 4226   × cxp 5248  ‘cfv 6030  2^nd c2nd 7318  .efcedgf 26088  iEdgciedg 26096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-iedg 26098

This theorem is referenced by:  opiedgfv  26108