Definition df-iedg 28972

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 28970	. 2 class iEdg
2		vg	. . 3 setvar 𝑔
3		cvv 3436	. . 3 class V
4	2	cv 1540	. . . . 5 class 𝑔
5	3, 3	cxp 5609	. . . . 5 class (V × V)
6	4, 5	wcel 2111	. . . 4 wff 𝑔 ∈ (V × V)
7		c2nd 7915	. . . . 5 class 2nd
8	4, 7	cfv 6476	. . . 4 class (2nd ‘𝑔)
9		cedgf 28961	. . . . 5 class .ef
10	4, 9	cfv 6476	. . . 4 class (.ef‘𝑔)
11	6, 8, 10	cif 4470	. . 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 1541	1 wff iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))

This definition is referenced by:  iedgval  28974