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Theorem qsinxp 6983
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 6982 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  x  e.  A )  ->  [ x ] R  =  [ x ] ( R  i^i  ( A  X.  A ) ) )
21eqeq2d 2449 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  x  e.  A )  ->  ( y  =  [
x ] R  <->  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) ) )
32rexbidva 2724 . . 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 2552 . 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 6914 . 2  |-  ( A /. R )  =  { y  |  E. x  e.  A  y  =  [ x ] R }
6 df-qs 6914 . 2  |-  ( A /. ( R  i^i  ( A  X.  A
) ) )  =  { y  |  E. x  e.  A  y  =  [ x ] ( R  i^i  ( A  X.  A ) ) }
74, 5, 63eqtr4g 2495 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 360    = wceq 1653    e. wcel 1726   {cab 2424   E.wrex 2708    i^i cin 3321    C_ wss 3322    X. cxp 4879   "cima 4884   [cec 6906   /.cqs 6907
This theorem is referenced by:  pi1buni  19070  pi1bas3  19073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-ec 6910  df-qs 6914
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