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Theorem vrgpfval 16910
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 3118 . . 3  |-  ( I  e.  V  ->  I  e.  _V )
3 id 22 . . . . 5  |-  ( i  =  I  ->  i  =  I )
4 fveq2 5872 . . . . . . 7  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  ( ~FG  `  I
) )
5 vrgpfval.r . . . . . . 7  |-  .~  =  ( ~FG  `  I )
64, 5syl6eqr 2516 . . . . . 6  |-  ( i  =  I  ->  ( ~FG  `  i
)  =  .~  )
7 eceq2 7367 . . . . . 6  |-  ( ( ~FG  `  i )  =  .~  ->  [ <" <. j ,  (/) >. "> ] ( ~FG  `  i )  =  [ <" <. j ,  (/) >. "> ]  .~  )
86, 7syl 16 . . . . 5  |-  ( i  =  I  ->  [ <"
<. j ,  (/) >. "> ] ( ~FG  `  i )  =  [ <" <. j ,  (/) >. "> ]  .~  )
93, 8mpteq12dv 4535 . . . 4  |-  ( i  =  I  ->  (
j  e.  i  |->  [
<" <. j ,  (/) >. "> ] ( ~FG  `  i
) )  =  ( j  e.  I  |->  [
<" <. j ,  (/) >. "> ]  .~  )
)
10 df-vrgp 16855 . . . 4  |- varFGrp  =  ( i  e. 
_V  |->  ( j  e.  i  |->  [ <" <. j ,  (/) >. "> ] ( ~FG  `  i ) ) )
11 vex 3112 . . . . 5  |-  i  e. 
_V
1211mptex 6144 . . . 4  |-  ( j  e.  i  |->  [ <"
<. j ,  (/) >. "> ] ( ~FG  `  i ) )  e. 
_V
139, 10, 12fvmpt3i 5960 . . 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 2510 1  |-  ( I  e.  V  ->  U  =  ( j  e.  I  |->  [ <" <. j ,  (/) >. "> ]  .~  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   <.cop 4038    |-> cmpt 4515   ` cfv 5594   [cec 7327   <"cs1 12540   ~FG cefg 16850  varFGrpcvrgp 16852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ec 7331  df-vrgp 16855
This theorem is referenced by:  vrgpval  16911  vrgpf  16912
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