Definition df-iedg 29082

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 29080	. 2 class iEdg
2		vg	. . 3 setvar 𝑔
3		cvv 3430	. . 3 class V
4	2	cv 1541	. . . . 5 class 𝑔
5	3, 3	cxp 5622	. . . . 5 class (V × V)
6	4, 5	wcel 2114	. . . 4 wff 𝑔 ∈ (V × V)
7		c2nd 7934	. . . . 5 class 2nd
8	4, 7	cfv 6492	. . . 4 class (2nd ‘𝑔)
9		cedgf 29071	. . . . 5 class .ef
10	4, 9	cfv 6492	. . . 4 class (.ef‘𝑔)
11	6, 8, 10	cif 4467	. . 3 class if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔))
12	2, 3, 11	cmpt 5167	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
13	1, 12	wceq 1542	1 wff iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))

This definition is referenced by:  iedgval  29084