Theorem vtxvalg 16011

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 16009	. 2 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
2		eleq1 2295	. . 3 |- ( g = G -> ( g e. ( _V X. _V ) <-> G e. ( _V X. _V ) ) )
3		fveq2 5670	. . 3 |- ( g = G -> ( 1st ` g ) = ( 1st ` G ) )
4		fveq2 5670	. . 3 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
5	2, 3, 4	ifbieq12d 3649	. 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 2825	. 2 |- ( G e. V -> G e. _V )
7		1stexg 6361	. . 3 |- ( G e. V -> ( 1st ` G ) e. _V )
8		basfn 13271	. . . 4 |- Base Fn _V
9		funfvex 5687	. . . . 5 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
10	9	funfni 5458	. . . 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 4605	. 2 |- ( G e. V -> if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) e. _V )
13	1, 5, 6, 12	fvmptd3 5771	1 |- ( G e. V -> (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   ifcif 3620   X. cxp 4747   Fn wfn 5347   ` cfv 5352   1stc1st 6332   Basecbs 13212  Vtxcvtx 16007

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-1st 6334  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-vtx 16009

This theorem is referenced by:  vtxex  16013  opvtxval  16016  funvtxdm2domval  16024  funvtxdm2vald  16026  vtxval0  16048