Definition df-vtx 15836

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 15834	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2799	. . 3 class V
4	2	cv 1394	. . . . 5 class 𝑔
5	3, 3	cxp 4718	. . . . 5 class (V × V)
6	4, 5	wcel 2200	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6293	. . . . 5 class 1st
8	4, 7	cfv 5321	. . . 4 class (1st ‘𝑔)
9		cbs 13053	. . . . 5 class Base
10	4, 9	cfv 5321	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3602	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4145	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1395	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxvalg  15838  1vgrex  15842  wlkreslem  16148