Definition df-vtx 16460

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 16458	. 2 class Vtx
2		vg	. . 3 setvar 𝑔
3		cvv 2821	. . 3 class V
4	2	cv 1401	. . . . 5 class 𝑔
5	3, 3	cxp 4774	. . . . 5 class (V × V)
6	4, 5	wcel 2209	. . . 4 wff 𝑔 ∈ (V × V)
7		c1st 6376	. . . . 5 class 1st
8	4, 7	cfv 5380	. . . 4 class (1st ‘𝑔)
9		cbs 13410	. . . . 5 class Base
10	4, 9	cfv 5380	. . . 4 class (Base‘𝑔)
11	6, 8, 10	cif 3638	. . 3 class if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔))
12	2, 3, 11	cmpt 4193	. 2 class (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
13	1, 12	wceq 1402	1 wff Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))

This definition is used by:  vtxvalg  16462  1vgrex  16466  wlkreslem  16824  clwwlknonmpo  16874  trlsegvdegfi  16913  eupth2lem3lem1fi  16914  eupth2lem3lem2fi  16915  eupth2lem3lem6fi  16917  eupth2lem3lem4fi  16919