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Theorem vrgpfval 16998
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 3065 . . 3  |-  ( I  e.  V  ->  I  e.  _V )
3 id 22 . . . . 5  |-  ( i  =  I  ->  i  =  I )
4 fveq2 5803 . . . . . . 7  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  ( ~FG  `  I
) )
5 vrgpfval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
64, 5syl6eqr 2459 . . . . . 6  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  .~  )
7 eceq2 7304 . . . . . 6  |-  ( ( ~FG  `  i )  =  .~  ->  [ <" <. j ,  (/) >. "> ] ( ~FG  `  i )  =  [ <" <. j ,  (/) >. "> ]  .~  )
86, 7syl 17 . . . . 5  |-  ( i  =  I  ->  [ <"
<. j ,  (/) >. "> ] ( ~FG  `  i )  =  [ <" <. j ,  (/) >. "> ]  .~  )
93, 8mpteq12dv 4470 . . . 4  |-  ( i  =  I  ->  (
j  e.  i  |->  [
<" <. j ,  (/) >. "> ] ( ~FG  `  i
) )  =  ( j  e.  I  |->  [
<" <. j ,  (/) >. "> ]  .~  )
)
10 df-vrgp 16943 . . . 4  |- varFGrp  =  ( i  e. 
_V  |->  ( j  e.  i  |->  [ <" <. j ,  (/) >. "> ] ( ~FG  `  i ) ) )
11 vex 3059 . . . . 5  |-  i  e. 
_V
1211mptex 6078 . . . 4  |-  ( j  e.  i  |->  [ <"
<. j ,  (/) >. "> ] ( ~FG  `  i ) )  e. 
_V
139, 10, 12fvmpt3i 5891 . . 3  |-  ( I  e.  _V  ->  (varFGrp `  I
)  =  ( j  e.  I  |->  [ <"
<. j ,  (/) >. "> ]  .~  ) )
142, 13syl 17 . 2  |-  ( I  e.  V  ->  (varFGrp `  I
)  =  ( j  e.  I  |->  [ <"
<. j ,  (/) >. "> ]  .~  ) )
151, 14syl5eq 2453 1  |-  ( I  e.  V  ->  U  =  ( j  e.  I  |->  [ <" <. j ,  (/) >. "> ]  .~  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840   _Vcvv 3056   (/)c0 3735   <.cop 3975    |-> cmpt 4450   ` cfv 5523   [cec 7264   <"cs1 12491   ~FG cefg 16938  varFGrpcvrgp 16940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ec 7268  df-vrgp 16943
This theorem is referenced by:  vrgpval  16999  vrgpf  17000
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