Definition df-vtx 29606

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 29604	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3454	. . 3 class V
4	2	cv 1569	. . . . 5 class 𝑔
5	3, 3	cxp 5653	. . . . 5 class (V × V)
6	4, 5	wcel 2147	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 8023	. . . . 5 class 1st
8	4, 7	cfv 6558	. . . 4 class (1st ‘𝑔)
9		cbs 17416	. . . . 5 class Base
10	4, 9	cfv 6558	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4485	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5189	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1570	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is used by:  vtxval  29608