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Theorem qsinxp 7405
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 7404 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  x  e.  A )  ->  [ x ] R  =  [ x ] ( R  i^i  ( A  X.  A ) ) )
21eqeq2d 2471 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  x  e.  A )  ->  ( y  =  [
x ] R  <->  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) ) )
32rexbidva 2965 . . 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 2593 . 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 7335 . 2  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
6 df-qs 7335 . 2  |-  ( A /. ( R  i^i  ( A  X.  A
) ) )  =  { y  |  E. x  e.  A  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) }
74, 5, 63eqtr4g 2523 1  |-  ( ( R " A ) 
C_  A  ->  ( A /. R )  =  ( A /. ( R  i^i  ( A  X.  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   E.wrex 2808    i^i cin 3470    C_ wss 3471    X. cxp 5006   "cima 5011   [cec 7327   /.cqs 7328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-ec 7331  df-qs 7335
This theorem is referenced by:  pi1buni  21666  pi1bas3  21669
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