Theorem vtxvalsnop 25933

Description: Degenerated case 2 for vertices: The set of vertices of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)

Hypotheses

Ref	Expression
vtxvalsnop.b	|- B e. _V
vtxvalsnop.g	|- G = { <. B , B >. }

Assertion

Ref	Expression
vtxvalsnop	|- (Vtx ` G ) = { B }

Proof of Theorem vtxvalsnop

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- B = B
2		vtxvalsnop.b	. . 3 |- B e. _V
3		vtxvalsnop.g	. . 3 |- G = { <. B , B >. }
4	2, 2, 3	funsneqopsn 6417	. 2 |- ( B = B -> G = <. { B } , { B } >. )
5		fveq2 6191	. . 3 |- ( G = <. { B } , { B } >. -> (Vtx ` G ) = (Vtx ` <. { B } , { B } >. ) )
6		snex 4908	. . . 4 |- { B } e. _V
7	6, 6	opvtxfvi 25889	. . 3 |- (Vtx ` <. { B } , { B } >. ) = { B }
8	5, 7	syl6eq 2672	. 2 |- ( G = <. { B } , { B } >. -> (Vtx ` G ) = { B } )
9	1, 4, 8	mp2b 10	1 |- (Vtx ` G ) = { B }

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   ` cfv 5888  Vtxcvtx 25874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-vtx 25876

This theorem is referenced by:  vtxval3sn  25935