Definition df-vtx 27368

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 27366	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3432	. . 3 class V
4	2	cv 1538	. . . . 5 class 𝑔
5	3, 3	cxp 5587	. . . . 5 class (V × V)
6	4, 5	wcel 2106	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7829	. . . . 5 class 1st
8	4, 7	cfv 6433	. . . 4 class (1st ‘𝑔)
9		cbs 16912	. . . . 5 class Base
10	4, 9	cfv 6433	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4459	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5157	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1539	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  27370