Theorem vtxval 25878

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

Assertion

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

Proof of Theorem vtxval

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . 4 |- ( g = G -> ( g e. ( _V X. _V ) <-> G e. ( _V X. _V ) ) )
2		fveq2 6191	. . . 4 |- ( g = G -> ( 1st ` g ) = ( 1st ` G ) )
3		fveq2 6191	. . . 4 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4	1, 2, 3	ifbieq12d 4113	. . 3 |- ( g = G -> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )
5		df-vtx 25876	. . 3 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
6		fvex 6201	. . . 4 |- ( 1st ` G ) e. _V
7		fvex 6201	. . . 4 |- ( Base ` G ) e. _V
8	6, 7	ifex 4156	. . 3 |- if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) e. _V
9	4, 5, 8	fvmpt 6282	. 2 |- ( G e. _V -> (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )
10		fvprc 6185	. . 3 |- ( -. G e. _V -> ( Base ` G ) = (/) )
11		prcnel 3218	. . . 4 |- ( -. G e. _V -> -. G e. ( _V X. _V ) )
12	11	iffalsed 4097	. . 3 |- ( -. G e. _V -> if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) = ( Base ` G ) )
13		fvprc 6185	. . 3 |- ( -. G e. _V -> (Vtx ` G ) = (/) )
14	10, 12, 13	3eqtr4rd 2667	. 2 |- ( -. G e. _V -> (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )
15	9, 14	pm2.61i 176	1 |- (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   X. cxp 5112   ` cfv 5888   1stc1st 7166   Basecbs 15857  Vtxcvtx 25874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-vtx 25876

This theorem is referenced by:  opvtxval  25883  funvtxdmge2val  25891  funvtxdm2val  25893  snstrvtxval  25929  vtxval0  25931