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Theorem vrgpfval 16261
Description: The canonical injection from the generating set  I to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
vrgpfval.r  |-  .~  =  ( ~FG  `  I )
vrgpfval.u  |-  U  =  (varFGrp `  I )
Assertion
Ref Expression
vrgpfval  |-  ( I  e.  V  ->  U  =  ( j  e.  I  |->  [ <" <. j ,  (/) >. "> ]  .~  ) )
Distinct variable groups:    j, I    .~ , j    j, V
Allowed substitution hint:    U( j)

Proof of Theorem vrgpfval
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 vrgpfval.u . 2  |-  U  =  (varFGrp `  I )
2 elex 2979 . . 3  |-  ( I  e.  V  ->  I  e.  _V )
3 id 22 . . . . 5  |-  ( i  =  I  ->  i  =  I )
4 fveq2 5689 . . . . . . 7  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  ( ~FG  `  I
) )
5 vrgpfval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
64, 5syl6eqr 2491 . . . . . 6  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  .~  )
7 eceq2 7136 . . . . . 6  |-  ( ( ~FG  `  i )  =  .~  ->  [ <" <. j ,  (/) >. "> ] ( ~FG  `  i )  =  [ <" <. j ,  (/) >. "> ]  .~  )
86, 7syl 16 . . . . 5  |-  ( i  =  I  ->  [ <"
<. j ,  (/) >. "> ] ( ~FG  `  i )  =  [ <" <. j ,  (/) >. "> ]  .~  )
93, 8mpteq12dv 4368 . . . 4  |-  ( i  =  I  ->  (
j  e.  i  |->  [
<" <. j ,  (/) >. "> ] ( ~FG  `  i
) )  =  ( j  e.  I  |->  [
<" <. j ,  (/) >. "> ]  .~  )
)
10 df-vrgp 16206 . . . 4  |- varFGrp  =  ( i  e. 
_V  |->  ( j  e.  i  |->  [ <" <. j ,  (/) >. "> ] ( ~FG  `  i ) ) )
11 vex 2973 . . . . 5  |-  i  e. 
_V
1211mptex 5946 . . . 4  |-  ( j  e.  i  |->  [ <"
<. j ,  (/) >. "> ] ( ~FG  `  i ) )  e. 
_V
139, 10, 12fvmpt3i 5776 . . 3  |-  ( I  e.  _V  ->  (varFGrp `  I
)  =  ( j  e.  I  |->  [ <"
<. j ,  (/) >. "> ]  .~  ) )
142, 13syl 16 . 2  |-  ( I  e.  V  ->  (varFGrp `  I
)  =  ( j  e.  I  |->  [ <"
<. j ,  (/) >. "> ]  .~  ) )
151, 14syl5eq 2485 1  |-  ( I  e.  V  ->  U  =  ( j  e.  I  |->  [ <" <. j ,  (/) >. "> ]  .~  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2970   (/)c0 3635   <.cop 3881    e. cmpt 4348   ` cfv 5416   [cec 7097   <"cs1 12222   ~FG cefg 16201  varFGrpcvrgp 16203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ec 7101  df-vrgp 16206
This theorem is referenced by:  vrgpval  16262  vrgpf  16263
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