Definition df-vtx 29581

Description: Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)

Assertion

Ref	Expression
df-vtx	⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

Detailed syntax breakdown of Definition df-vtx

Step	Hyp	Ref	Expression
1		cvtx 29579	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3450	. . 3 class V
4	2	cv 1568	. . . . 5 class 𝑔
5	3, 3	cxp 5648	. . . . 5 class (V × V)
6	4, 5	wcel 2144	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7999	. . . . 5 class 1st
8	4, 7	cfv 6538	. . . 4 class (1st ‘𝑔)
9		cbs 17392	. . . . 5 class Base
10	4, 9	cfv 6538	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4481	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5185	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1569	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is used by:  vtxval  29583