Definition df-iedg 26784

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 26782	. 2 class iEdg
2		vg	. . 3 setvar 𝑔
3		cvv 3494	. . 3 class V
4	2	cv 1536	. . . . 5 class 𝑔
5	3, 3	cxp 5553	. . . . 5 class (V × V)
6	4, 5	wcel 2114	. . . 4 wff 𝑔 ∈ (V × V)
7		c2nd 7688	. . . . 5 class 2nd
8	4, 7	cfv 6355	. . . 4 class (2nd ‘𝑔)
9		cedgf 26774	. . . . 5 class .ef
10	4, 9	cfv 6355	. . . 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 5146	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
13	1, 12	wceq 1537	1 wff iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))

This definition is referenced by:  iedgval  26786