Definition df-vtx 15864

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 15862	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2802	. . 3 class V
4	2	cv 1396	. . . . 5 class 𝑔
5	3, 3	cxp 4723	. . . . 5 class (V × V)
6	4, 5	wcel 2202	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6300	. . . . 5 class 1st
8	4, 7	cfv 5326	. . . 4 class (1st ‘𝑔)
9		cbs 13081	. . . . 5 class Base
10	4, 9	cfv 5326	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3605	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4150	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1397	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is referenced by:  vtxvalg  15866  1vgrex  15870  wlkreslem  16228  clwwlknonmpo  16278