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Theorem vrgpval 16365
Description: The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
vrgpfval.r  |-  .~  =  ( ~FG  `  I )
vrgpfval.u  |-  U  =  (varFGrp `  I )
Assertion
Ref Expression
vrgpval  |-  ( ( I  e.  V  /\  A  e.  I )  ->  ( U `  A
)  =  [ <"
<. A ,  (/) >. "> ]  .~  )

Proof of Theorem vrgpval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 vrgpfval.r . . . 4  |-  .~  =  ( ~FG  `  I )
2 vrgpfval.u . . . 4  |-  U  =  (varFGrp `  I )
31, 2vrgpfval 16364 . . 3  |-  ( I  e.  V  ->  U  =  ( j  e.  I  |->  [ <" <. j ,  (/) >. "> ]  .~  ) )
43fveq1d 5788 . 2  |-  ( I  e.  V  ->  ( U `  A )  =  ( ( j  e.  I  |->  [ <"
<. j ,  (/) >. "> ]  .~  ) `  A
) )
5 opeq1 4154 . . . . 5  |-  ( j  =  A  ->  <. j ,  (/) >.  =  <. A ,  (/) >. )
65s1eqd 12391 . . . 4  |-  ( j  =  A  ->  <" <. j ,  (/) >. ">  =  <" <. A ,  (/) >. "> )
7 eceq1 7234 . . . 4  |-  ( <" <. j ,  (/) >. ">  =  <" <. A ,  (/) >. ">  ->  [
<" <. j ,  (/) >. "> ]  .~  =  [ <" <. A ,  (/)
>. "> ]  .~  )
86, 7syl 16 . . 3  |-  ( j  =  A  ->  [ <"
<. j ,  (/) >. "> ]  .~  =  [ <"
<. A ,  (/) >. "> ]  .~  )
9 eqid 2451 . . 3  |-  ( j  e.  I  |->  [ <"
<. j ,  (/) >. "> ]  .~  )  =  ( j  e.  I  |->  [
<" <. j ,  (/) >. "> ]  .~  )
10 fvex 5796 . . . . 5  |-  ( ~FG  `  I
)  e.  _V
111, 10eqeltri 2533 . . . 4  |-  .~  e.  _V
12 ecexg 7202 . . . 4  |-  (  .~  e.  _V  ->  [ <" <. A ,  (/) >. "> ]  .~  e.  _V )
1311, 12ax-mp 5 . . 3  |-  [ <"
<. A ,  (/) >. "> ]  .~  e.  _V
148, 9, 13fvmpt 5870 . 2  |-  ( A  e.  I  ->  (
( j  e.  I  |->  [ <" <. j ,  (/) >. "> ]  .~  ) `  A )  =  [ <" <. A ,  (/) >. "> ]  .~  )
154, 14sylan9eq 2511 1  |-  ( ( I  e.  V  /\  A  e.  I )  ->  ( U `  A
)  =  [ <"
<. A ,  (/) >. "> ]  .~  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3065   (/)c0 3732   <.cop 3978    |-> cmpt 4445   ` cfv 5513   [cec 7196   <"cs1 12323   ~FG cefg 16304  varFGrpcvrgp 16306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ec 7200  df-s1 12331  df-vrgp 16309
This theorem is referenced by:  vrgpinv  16367  frgpup2  16374  frgpup3lem  16375  frgpnabllem1  16452
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