Theorem vtxvalg 15857

Description: The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)

Assertion

Ref	Expression
vtxvalg	|- ( G e. V -> (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )

Proof of Theorem vtxvalg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-vtx 15855	. 2 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
2		eleq1 2292	. . 3 |- ( g = G -> ( g e. ( _V X. _V ) <-> G e. ( _V X. _V ) ) )
3		fveq2 5635	. . 3 |- ( g = G -> ( 1st ` g ) = ( 1st ` G ) )
4		fveq2 5635	. . 3 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
5	2, 3, 4	ifbieq12d 3630	. 2 |- ( g = G -> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )
6		elex 2812	. 2 |- ( G e. V -> G e. _V )
7		1stexg 6325	. . 3 |- ( G e. V -> ( 1st ` G ) e. _V )
8		basfn 13131	. . . 4 |- Base Fn _V
9		funfvex 5652	. . . . 5 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
10	9	funfni 5429	. . . 4 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
11	8, 6, 10	sylancr 414	. . 3 |- ( G e. V -> ( Base ` G ) e. _V )
12	7, 11	ifexd 4579	. 2 |- ( G e. V -> if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) e. _V )
13	1, 5, 6, 12	fvmptd3 5736	1 |- ( G e. V -> (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2800   ifcif 3603   X. cxp 4721   Fn wfn 5319   ` cfv 5324   1stc1st 6296   Basecbs 13072  Vtxcvtx 15853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fo 5330  df-fv 5332  df-1st 6298  df-inn 9134  df-ndx 13075  df-slot 13076  df-base 13078  df-vtx 15855

This theorem is referenced by:  vtxex  15859  opvtxval  15862  funvtxdm2domval  15870  funvtxdm2vald  15872  vtxval0  15894