Definition df-iedg 29197

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 29195	. 2 class iEdg
2		vg	. . 3 setvar 𝑔
3		cvv 3454	. . 3 class V
4	2	cv 1559	. . . . 5 class 𝑔
5	3, 3	cxp 5645	. . . . 5 class (V × V)
6	4, 5	wcel 2142	. . . 4 wff 𝑔 ∈ (V × V)
7		c2nd 7969	. . . . 5 class 2nd
8	4, 7	cfv 6521	. . . 4 class (2nd ‘𝑔)
9		cedgf 29186	. . . . 5 class .ef
10	4, 9	cfv 6521	. . . 4 class (.ef‘𝑔)
11	6, 8, 10	cif 4480	. . 3 class if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔))
12	2, 3, 11	cmpt 5181	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
13	1, 12	wceq 1560	1 wff iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))

This definition is referenced by:  iedgval  29199