Definition df-iedg 27357

Description: Define the function mapping a graph to its indexed edges. This definition is very general: It defines the indexed edges for any ordered pair as its second component, and for any other class as its "edge function". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure (containing a slot for "edge functions") representing a graph. (Contributed by AV, 20-Sep-2020.)

Assertion

Ref	Expression
df-iedg	⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))

Detailed syntax breakdown of Definition df-iedg

Step	Hyp	Ref	Expression
1		ciedg 27355	. 2 class iEdg
2		vg	. . 3 setvar 𝑔
3		cvv 3430	. . 3 class V
4	2	cv 1538	. . . . 5 class 𝑔
5	3, 3	cxp 5583	. . . . 5 class (V × V)
6	4, 5	wcel 2106	. . . 4 wff 𝑔 ∈ (V × V)
7		c2nd 7820	. . . . 5 class 2nd
8	4, 7	cfv 6427	. . . 4 class (2nd ‘𝑔)
9		cedgf 27344	. . . . 5 class .ef
10	4, 9	cfv 6427	. . . 4 class (.ef‘𝑔)
11	6, 8, 10	cif 4460	. . 3 class if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔))
12	2, 3, 11	cmpt 5157	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
13	1, 12	wceq 1539	1 wff iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))

This definition is referenced by:  iedgval  27359