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Theorem qsinxp 6558
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 6557 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  x  e.  A )  ->  [ x ] R  =  [ x ] ( R  i^i  ( A  X.  A ) ) )
21eqeq2d 2169 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  x  e.  A )  ->  ( y  =  [
x ] R  <->  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) ) )
32rexbidva 2454 . . 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 2275 . 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 6488 . 2  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
6 df-qs 6488 . 2  |-  ( A /. ( R  i^i  ( A  X.  A
) ) )  =  { y  |  E. x  e.  A  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) }
74, 5, 63eqtr4g 2215 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 103    = wceq 1335    e. wcel 2128   {cab 2143   E.wrex 2436    i^i cin 3101    C_ wss 3102    X. cxp 4586   "cima 4591   [cec 6480   /.cqs 6481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-ec 6484  df-qs 6488
This theorem is referenced by: (None)
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