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Theorem divsin 13771
Description: Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
divsin.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
divsin.v  |-  ( ph  ->  V  =  ( Base `  R ) )
divsin.e  |-  ( ph  ->  .~  e.  W )
divsin.r  |-  ( ph  ->  R  e.  Z )
divsin.s  |-  ( ph  ->  (  .~  " V
)  C_  V )
Assertion
Ref Expression
divsin  |-  ( ph  ->  U  =  ( R 
/.s  (  .~  i^i  ( V  X.  V ) ) ) )

Proof of Theorem divsin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 divsin.s . . . . 5  |-  ( ph  ->  (  .~  " V
)  C_  V )
2 ecinxp 6981 . . . . 5  |-  ( ( (  .~  " V
)  C_  V  /\  x  e.  V )  ->  [ x ]  .~  =  [ x ] (  .~  i^i  ( V  X.  V ) ) )
31, 2sylan 459 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [ x ]  .~  =  [ x ] (  .~  i^i  ( V  X.  V
) ) )
43mpteq2dva 4297 . . 3  |-  ( ph  ->  ( x  e.  V  |->  [ x ]  .~  )  =  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V
) ) ) )
54oveq1d 6098 . 2  |-  ( ph  ->  ( ( x  e.  V  |->  [ x ]  .~  )  "s  R )  =  ( ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V ) ) )  "s  R ) )
6 divsin.u . . 3  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
7 divsin.v . . 3  |-  ( ph  ->  V  =  ( Base `  R ) )
8 eqid 2438 . . 3  |-  ( x  e.  V  |->  [ x ]  .~  )  =  ( x  e.  V  |->  [ x ]  .~  )
9 divsin.e . . 3  |-  ( ph  ->  .~  e.  W )
10 divsin.r . . 3  |-  ( ph  ->  R  e.  Z )
116, 7, 8, 9, 10divsval 13769 . 2  |-  ( ph  ->  U  =  ( ( x  e.  V  |->  [ x ]  .~  )  "s  R ) )
12 eqidd 2439 . . 3  |-  ( ph  ->  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) )  =  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) ) )
13 eqid 2438 . . 3  |-  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V
) ) )  =  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V ) ) )
14 inex1g 4348 . . . 4  |-  (  .~  e.  W  ->  (  .~  i^i  ( V  X.  V
) )  e.  _V )
159, 14syl 16 . . 3  |-  ( ph  ->  (  .~  i^i  ( V  X.  V ) )  e.  _V )
1612, 7, 13, 15, 10divsval 13769 . 2  |-  ( ph  ->  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) )  =  ( ( x  e.  V  |->  [ x ]
(  .~  i^i  ( V  X.  V ) ) )  "s  R ) )
175, 11, 163eqtr4d 2480 1  |-  ( ph  ->  U  =  ( R 
/.s  (  .~  i^i  ( V  X.  V ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321    C_ wss 3322    e. cmpt 4268    X. cxp 4878   "cima 4883   ` cfv 5456  (class class class)co 6083   [cec 6905   Basecbs 13471    "s cimas 13732    /.s cqus 13733
This theorem is referenced by:  pi1addf  19074  pi1addval  19075  pi1grplem  19076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-ec 6909  df-divs 13737
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