Definition df-iedg 25877

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 25875	. 2 class iEdg
2		vg	. . 3 setvar 𝑔
3		cvv 3200	. . 3 class V
4	2	cv 1482	. . . . 5 class 𝑔
5	3, 3	cxp 5112	. . . . 5 class (V × V)
6	4, 5	wcel 1990	. . . 4 wff 𝑔 ∈ (V × V)
7		c2nd 7167	. . . . 5 class 2nd
8	4, 7	cfv 5888	. . . 4 class (2nd ‘𝑔)
9		cedgf 25867	. . . . 5 class .ef
10	4, 9	cfv 5888	. . . 4 class (.ef‘𝑔)
11	6, 8, 10	cif 4086	. . 3 class if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔))
12	2, 3, 11	cmpt 4729	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
13	1, 12	wceq 1483	1 wff iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))

This definition is referenced by:  iedgval  25879  iedgvalOLD  25881