Definition df-vtx 28255

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 28253	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3474	. . 3 class V
4	2	cv 1540	. . . . 5 class 𝑔
5	3, 3	cxp 5674	. . . . 5 class (V × V)
6	4, 5	wcel 2106	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7972	. . . . 5 class 1st
8	4, 7	cfv 6543	. . . 4 class (1st ‘𝑔)
9		cbs 17143	. . . . 5 class Base
10	4, 9	cfv 6543	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4528	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5231	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1541	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxval  28257