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Theorem divsin 13446
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 6734 . . . . 5  |-  ( ( (  .~  " V
)  C_  V  /\  x  e.  V )  ->  [ x ]  .~  =  [ x ] (  .~  i^i  ( V  X.  V ) ) )
31, 2sylan 457 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [ x ]  .~  =  [ x ] (  .~  i^i  ( V  X.  V
) ) )
43mpteq2dva 4106 . . 3  |-  ( ph  ->  ( x  e.  V  |->  [ x ]  .~  )  =  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V
) ) ) )
54oveq1d 5873 . 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 2283 . . 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 13444 . 2  |-  ( ph  ->  U  =  ( ( x  e.  V  |->  [ x ]  .~  )  "s  R ) )
12 eqidd 2284 . . 3  |-  ( ph  ->  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) )  =  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) ) )
13 eqid 2283 . . 3  |-  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V
) ) )  =  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V ) ) )
14 inex1g 4157 . . . 4  |-  (  .~  e.  W  ->  (  .~  i^i  ( V  X.  V
) )  e.  _V )
159, 14syl 15 . . 3  |-  ( ph  ->  (  .~  i^i  ( V  X.  V ) )  e.  _V )
1612, 7, 13, 15, 10divsval 13444 . 2  |-  ( ph  ->  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) )  =  ( ( x  e.  V  |->  [ x ]
(  .~  i^i  ( V  X.  V ) ) )  "s  R ) )
175, 11, 163eqtr4d 2325 1  |-  ( ph  ->  U  =  ( R 
/.s  (  .~  i^i  ( V  X.  V ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152    e. cmpt 4077    X. cxp 4687   "cima 4692   ` cfv 5255  (class class class)co 5858   [cec 6658   Basecbs 13148    "s cimas 13407    /.s cqus 13408
This theorem is referenced by:  pi1addf  18545  pi1addval  18546  pi1grplem  18547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ec 6662  df-divs 13412
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