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Theorem qsinxp 6368
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 6367 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  x  e.  A )  ->  [ x ] R  =  [ x ] ( R  i^i  ( A  X.  A ) ) )
21eqeq2d 2099 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  x  e.  A )  ->  ( y  =  [
x ] R  <->  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) ) )
32rexbidva 2377 . . 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 2205 . 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 6298 . 2  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
6 df-qs 6298 . 2  |-  ( A /. ( R  i^i  ( A  X.  A
) ) )  =  { y  |  E. x  e.  A  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) }
74, 5, 63eqtr4g 2145 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 102    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360    i^i cin 2998    C_ wss 2999    X. cxp 4436   "cima 4441   [cec 6290   /.cqs 6291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-ec 6294  df-qs 6298
This theorem is referenced by: (None)
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