Definition df-vtx 15688

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 15686	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2773	. . 3 class V
4	2	cv 1372	. . . . 5 class 𝑔
5	3, 3	cxp 4681	. . . . 5 class (V × V)
6	4, 5	wcel 2177	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6237	. . . . 5 class 1st
8	4, 7	cfv 5280	. . . 4 class (1st ‘𝑔)
9		cbs 12907	. . . . 5 class Base
10	4, 9	cfv 5280	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3575	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4113	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1373	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxvalg  15690  1vgrex  15694