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Theorem divsin 14605
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 7288 . . . . 5  |-  ( ( (  .~  " V
)  C_  V  /\  x  e.  V )  ->  [ x ]  .~  =  [ x ] (  .~  i^i  ( V  X.  V ) ) )
31, 2sylan 471 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [ x ]  .~  =  [ x ] (  .~  i^i  ( V  X.  V
) ) )
43mpteq2dva 4489 . . 3  |-  ( ph  ->  ( x  e.  V  |->  [ x ]  .~  )  =  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V
) ) ) )
54oveq1d 6218 . 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 2454 . . 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 14603 . 2  |-  ( ph  ->  U  =  ( ( x  e.  V  |->  [ x ]  .~  )  "s  R ) )
12 eqidd 2455 . . 3  |-  ( ph  ->  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) )  =  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) ) )
13 eqid 2454 . . 3  |-  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V
) ) )  =  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V ) ) )
14 inex1g 4546 . . . 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 14603 . 2  |-  ( ph  ->  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) )  =  ( ( x  e.  V  |->  [ x ]
(  .~  i^i  ( V  X.  V ) ) )  "s  R ) )
175, 11, 163eqtr4d 2505 1  |-  ( ph  ->  U  =  ( R 
/.s  (  .~  i^i  ( V  X.  V ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078    i^i cin 3438    C_ wss 3439    |-> cmpt 4461    X. cxp 4949   "cima 4954   ` cfv 5529  (class class class)co 6203   [cec 7212   Basecbs 14296    "s cimas 14565    /.s cqus 14566
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-ec 7216  df-divs 14570
This theorem is referenced by:  pi1addf  20761  pi1addval  20762  pi1grplem  20763
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