Definition df-iedg 29570

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 29568	. 2 class iEdg
2		vg	. . 3 setvar 𝑔
3		cvv 3451	. . 3 class V
4	2	cv 1569	. . . . 5 class 𝑔
5	3, 3	cxp 5649	. . . . 5 class (V × V)
6	4, 5	wcel 2145	. . . 4 wff 𝑔 ∈ (V × V)
7		c2nd 7998	. . . . 5 class 2nd
8	4, 7	cfv 6537	. . . 4 class (2nd ‘𝑔)
9		cedgf 29559	. . . . 5 class .ef
10	4, 9	cfv 6537	. . . 4 class (.ef‘𝑔)
11	6, 8, 10	cif 4482	. . 3 class if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔))
12	2, 3, 11	cmpt 5186	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
13	1, 12	wceq 1570	1 wff iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))

This definition is used by:  iedgval  29572