Definition df-vtx 29385

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 29383	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 3458	. . 3 class V
4	2	cv 1569	. . . . 5 class 𝑔
5	3, 3	cxp 5664	. . . . 5 class (V × V)
6	4, 5	wcel 2146	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 7993	. . . . 5 class 1st
8	4, 7	cfv 6543	. . . 4 class (1st ‘𝑔)
9		cbs 17294	. . . . 5 class Base
10	4, 9	cfv 6543	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 4492	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 5197	. 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  29387