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Theorem qusin 15528
Description: Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
qusin.u  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
qusin.v  |-  ( ph  ->  V  =  ( Base `  R ) )
qusin.e  |-  ( ph  ->  .~  e.  W )
qusin.r  |-  ( ph  ->  R  e.  Z )
qusin.s  |-  ( ph  ->  (  .~  " V
)  C_  V )
Assertion
Ref Expression
qusin  |-  ( ph  ->  U  =  ( R 
/.s  (  .~  i^i  ( V  X.  V ) ) ) )

Proof of Theorem qusin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 qusin.s . . . . 5  |-  ( ph  ->  (  .~  " V
)  C_  V )
2 ecinxp 7456 . . . . 5  |-  ( ( (  .~  " V
)  C_  V  /\  x  e.  V )  ->  [ x ]  .~  =  [ x ] (  .~  i^i  ( V  X.  V ) ) )
31, 2sylan 479 . . . 4  |-  ( (
ph  /\  x  e.  V )  ->  [ x ]  .~  =  [ x ] (  .~  i^i  ( V  X.  V
) ) )
43mpteq2dva 4482 . . 3  |-  ( ph  ->  ( x  e.  V  |->  [ x ]  .~  )  =  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V
) ) ) )
54oveq1d 6323 . 2  |-  ( ph  ->  ( ( x  e.  V  |->  [ x ]  .~  )  "s  R )  =  ( ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V ) ) )  "s  R ) )
6 qusin.u . . 3  |-  ( ph  ->  U  =  ( R 
/.s  .~  ) )
7 qusin.v . . 3  |-  ( ph  ->  V  =  ( Base `  R ) )
8 eqid 2471 . . 3  |-  ( x  e.  V  |->  [ x ]  .~  )  =  ( x  e.  V  |->  [ x ]  .~  )
9 qusin.e . . 3  |-  ( ph  ->  .~  e.  W )
10 qusin.r . . 3  |-  ( ph  ->  R  e.  Z )
116, 7, 8, 9, 10qusval 15526 . 2  |-  ( ph  ->  U  =  ( ( x  e.  V  |->  [ x ]  .~  )  "s  R ) )
12 eqidd 2472 . . 3  |-  ( ph  ->  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) )  =  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) ) )
13 eqid 2471 . . 3  |-  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V
) ) )  =  ( x  e.  V  |->  [ x ] (  .~  i^i  ( V  X.  V ) ) )
14 inex1g 4539 . . . 4  |-  (  .~  e.  W  ->  (  .~  i^i  ( V  X.  V
) )  e.  _V )
159, 14syl 17 . . 3  |-  ( ph  ->  (  .~  i^i  ( V  X.  V ) )  e.  _V )
1612, 7, 13, 15, 10qusval 15526 . 2  |-  ( ph  ->  ( R  /.s  (  .~  i^i  ( V  X.  V
) ) )  =  ( ( x  e.  V  |->  [ x ]
(  .~  i^i  ( V  X.  V ) ) )  "s  R ) )
175, 11, 163eqtr4d 2515 1  |-  ( ph  ->  U  =  ( R 
/.s  (  .~  i^i  ( V  X.  V ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389    C_ wss 3390    |-> cmpt 4454    X. cxp 4837   "cima 4842   ` cfv 5589  (class class class)co 6308   [cec 7379   Basecbs 15199    "s cimas 15480    /.s cqus 15482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-ec 7383  df-qus 15487
This theorem is referenced by:  pi1addf  22156  pi1addval  22157  pi1grplem  22158
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