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Theorem vtxval 39255
Description: The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
Assertion
Ref Expression
vtxval  |-  ( G  e.  V  ->  (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.
StepHypRef Expression
1 elex 3040 . 2  |-  ( G  e.  V  ->  G  e.  _V )
2 eleq1 2537 . . . 4  |-  ( g  =  G  ->  (
g  e.  ( _V 
X.  _V )  <->  G  e.  ( _V  X.  _V )
) )
3 fveq2 5879 . . . 4  |-  ( g  =  G  ->  ( 1st `  g )  =  ( 1st `  G
) )
4 fveq2 5879 . . . 4  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
52, 3, 4ifbieq12d 3899 . . 3  |-  ( g  =  G  ->  if ( g  e.  ( _V  X.  _V ) ,  ( 1st `  g
) ,  ( Base `  g ) )  =  if ( G  e.  ( _V  X.  _V ) ,  ( 1st `  G ) ,  (
Base `  G )
) )
6 df-vtx 39253 . . 3  |- Vtx  =  ( g  e.  _V  |->  if ( g  e.  ( _V  X.  _V ) ,  ( 1st `  g
) ,  ( Base `  g ) ) )
7 fvex 5889 . . . 4  |-  ( 1st `  G )  e.  _V
8 fvex 5889 . . . 4  |-  ( Base `  G )  e.  _V
97, 8ifex 3940 . . 3  |-  if ( G  e.  ( _V 
X.  _V ) ,  ( 1st `  G ) ,  ( Base `  G
) )  e.  _V
105, 6, 9fvmpt 5963 . 2  |-  ( G  e.  _V  ->  (Vtx `  G )  =  if ( G  e.  ( _V  X.  _V ) ,  ( 1st `  G
) ,  ( Base `  G ) ) )
111, 10syl 17 1  |-  ( G  e.  V  ->  (Vtx `  G )  =  if ( G  e.  ( _V  X.  _V ) ,  ( 1st `  G
) ,  ( Base `  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   _Vcvv 3031   ifcif 3872    X. cxp 4837   ` cfv 5589   1stc1st 6810   Basecbs 15199  Vtxcvtx 39251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-vtx 39253
This theorem is referenced by:  opvtxval  39258  funvtxdm2val  39266  funvtxdmge2val  39269  snstrvtxval  39290  vtxval0  39292  vtxvalsnop  39294
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