Theorem vtxvalg 15690

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 15688	. 2 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
2		eleq1 2269	. . 3 |- ( g = G -> ( g e. ( _V X. _V ) <-> G e. ( _V X. _V ) ) )
3		fveq2 5589	. . 3 |- ( g = G -> ( 1st ` g ) = ( 1st ` G ) )
4		fveq2 5589	. . 3 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
5	2, 3, 4	ifbieq12d 3602	. 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 2785	. 2 |- ( G e. V -> G e. _V )
7		1stexg 6266	. . 3 |- ( G e. V -> ( 1st ` G ) e. _V )
8		basfn 12965	. . . 4 |- Base Fn _V
9		funfvex 5606	. . . . 5 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
10	9	funfni 5385	. . . 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 4539	. 2 |- ( G e. V -> if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) e. _V )
13	1, 5, 6, 12	fvmptd3 5686	1 |- ( G e. V -> (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   _Vcvv 2773   ifcif 3575   X. cxp 4681   Fn wfn 5275   ` cfv 5280   1stc1st 6237   Basecbs 12907  Vtxcvtx 15686

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fo 5286  df-fv 5288  df-1st 6239  df-inn 9057  df-ndx 12910  df-slot 12911  df-base 12913  df-vtx 15688

This theorem is referenced by:  vtxex  15692  opvtxval  15695  funvtxdm2domval  15703  funvtxdm2vald  15705  vtxval0  15725