Definition df-vtx 29257

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 29255	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3457	. . 3 class V
4	2	cv 1562	. . . . 5 class 𝑔
5	3, 3	cxp 5650	. . . . 5 class (V × V)
6	4, 5	wcel 2145	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7972	. . . . 5 class 1st
8	4, 7	cfv 6525	. . . 4 class (1st ‘𝑔)
9		cbs 17259	. . . . 5 class Base
10	4, 9	cfv 6525	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4483	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5186	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1563	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  29259