Definition df-iedg 29021

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 29019	. 2 class iEdg
2		vg	. . 3 setvar 𝑔
3		cvv 3438	. . 3 class V
4	2	cv 1540	. . . . 5 class 𝑔
5	3, 3	cxp 5620	. . . . 5 class (V × V)
6	4, 5	wcel 2113	. . . 4 wff 𝑔 ∈ (V × V)
7		c2nd 7930	. . . . 5 class 2nd
8	4, 7	cfv 6490	. . . 4 class (2nd ‘𝑔)
9		cedgf 29010	. . . . 5 class .ef
10	4, 9	cfv 6490	. . . 4 class (.ef‘𝑔)
11	6, 8, 10	cif 4477	. . 3 class if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔))
12	2, 3, 11	cmpt 5177	. 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  29023