Definition df-vtx 27271

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 27269	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3422	. . 3 class V
4	2	cv 1538	. . . . 5 class 𝑔
5	3, 3	cxp 5578	. . . . 5 class (V × V)
6	4, 5	wcel 2108	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7802	. . . . 5 class 1st
8	4, 7	cfv 6418	. . . 4 class (1st ‘𝑔)
9		cbs 16840	. . . . 5 class Base
10	4, 9	cfv 6418	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4456	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5153	. 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  27273