Theorem vtxvalg 15866

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 15864	. 2 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
2		eleq1 2294	. . 3 |- ( g = G -> ( g e. ( _V X. _V ) <-> G e. ( _V X. _V ) ) )
3		fveq2 5639	. . 3 |- ( g = G -> ( 1st ` g ) = ( 1st ` G ) )
4		fveq2 5639	. . 3 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
5	2, 3, 4	ifbieq12d 3632	. 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 2814	. 2 |- ( G e. V -> G e. _V )
7		1stexg 6329	. . 3 |- ( G e. V -> ( 1st ` G ) e. _V )
8		basfn 13140	. . . 4 |- Base Fn _V
9		funfvex 5656	. . . . 5 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
10	9	funfni 5432	. . . 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 4581	. 2 |- ( G e. V -> if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) e. _V )
13	1, 5, 6, 12	fvmptd3 5740	1 |- ( G e. V -> (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   ifcif 3605   X. cxp 4723   Fn wfn 5321   ` cfv 5326   1stc1st 6300   Basecbs 13081  Vtxcvtx 15862

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-1st 6302  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-vtx 15864

This theorem is referenced by:  vtxex  15868  opvtxval  15871  funvtxdm2domval  15879  funvtxdm2vald  15881  vtxval0  15903