Definition df-vtx 15555

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 15553	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2771	. . 3 class V
4	2	cv 1371	. . . . 5 class 𝑔
5	3, 3	cxp 4672	. . . . 5 class (V × V)
6	4, 5	wcel 2175	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6223	. . . . 5 class 1st
8	4, 7	cfv 5270	. . . 4 class (1st ‘𝑔)
9		cbs 12774	. . . . 5 class Base
10	4, 9	cfv 5270	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3570	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4104	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1372	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxvalg  15557  1vgrex  15559