Theorem vtxvalg 16257

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 16255	. 2 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
2		eleq1 2301	. . 3 |- ( g = G -> ( g e. ( _V X. _V ) <-> G e. ( _V X. _V ) ) )
3		fveq2 5695	. . 3 |- ( g = G -> ( 1st ` g ) = ( 1st ` G ) )
4		fveq2 5695	. . 3 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
5	2, 3, 4	ifbieq12d 3667	. 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 2833	. 2 |- ( G e. V -> G e. _V )
7		1stexg 6401	. . 3 |- ( G e. V -> ( 1st ` G ) e. _V )
8		basfn 13411	. . . 4 |- Base Fn _V
9		funfvex 5712	. . . . 5 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
10	9	funfni 5483	. . . 4 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
11	8, 6, 10	sylancr 418	. . 3 |- ( G e. V -> ( Base ` G ) e. _V )
12	7, 11	ifexd 4630	. 2 |- ( G e. V -> if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) e. _V )
13	1, 5, 6, 12	fvmptd3 5799	1 |- ( G e. V -> (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   ifcif 3638   X. cxp 4772   Fn wfn 5372   ` cfv 5377   1stc1st 6372   Basecbs 13352  Vtxcvtx 16253

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-1st 6374  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358  df-vtx 16255

This theorem is used by:  vtxex  16259  opvtxval  16262  funvtxdm2domval  16270  funvtxdm2vald  16272  vtxval0  16294