Definition df-iedg 16170

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 16168	. 2 class iEdg
2		vg	. . 3 setvar 𝑔
3		cvv 2821	. . 3 class V
4	2	cv 1401	. . . . 5 class 𝑔
5	3, 3	cxp 4767	. . . . 5 class (V × V)
6	4, 5	wcel 2209	. . . 4 wff 𝑔 ∈ (V × V)
7		c2nd 6363	. . . . 5 class 2nd
8	4, 7	cfv 5372	. . . 4 class (2nd ‘𝑔)
9		cedgf 16159	. . . . 5 class .ef
10	4, 9	cfv 5372	. . . 4 class (.ef‘𝑔)
11	6, 8, 10	cif 3635	. . 3 class if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔))
12	2, 3, 11	cmpt 4187	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
13	1, 12	wceq 1402	1 wff iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))

This definition is referenced by:  iedgvalg  16172  edgval  16215