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Theorem qsinxp 6735
Description: Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Assertion
Ref Expression
qsinxp  |-  ( ( R " A ) 
C_  A  ->  ( A /. R )  =  ( A /. ( R  i^i  ( A  X.  A ) ) ) )

Proof of Theorem qsinxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecinxp 6734 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  x  e.  A )  ->  [ x ] R  =  [ x ] ( R  i^i  ( A  X.  A ) ) )
21eqeq2d 2294 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  x  e.  A )  ->  ( y  =  [
x ] R  <->  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) ) )
32rexbidva 2560 . . 3  |-  ( ( R " A ) 
C_  A  ->  ( E. x  e.  A  y  =  [ x ] R  <->  E. x  e.  A  y  =  [ x ] ( R  i^i  ( A  X.  A
) ) ) )
43abbidv 2397 . 2  |-  ( ( R " A ) 
C_  A  ->  { y  |  E. x  e.  A  y  =  [
x ] R }  =  { y  |  E. x  e.  A  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) } )
5 df-qs 6666 . 2  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
6 df-qs 6666 . 2  |-  ( A /. ( R  i^i  ( A  X.  A
) ) )  =  { y  |  E. x  e.  A  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) }
74, 5, 63eqtr4g 2340 1  |-  ( ( R " A ) 
C_  A  ->  ( A /. R )  =  ( A /. ( R  i^i  ( A  X.  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    i^i cin 3151    C_ wss 3152    X. cxp 4687   "cima 4692   [cec 6658   /.cqs 6659
This theorem is referenced by:  pi1buni  18538  pi1bas3  18541
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-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-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662  df-qs 6666
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