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Theorem ecinxp 6497
Description: Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)
Assertion
Ref Expression
ecinxp  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  [ B ] R  =  [ B ] ( R  i^i  ( A  X.  A ) ) )

Proof of Theorem ecinxp
StepHypRef Expression
1 simpr 109 . . . . . . . 8  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  B  e.  A )
21snssd 3660 . . . . . . 7  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  { B }  C_  A )
3 df-ss 3079 . . . . . . 7  |-  ( { B }  C_  A  <->  ( { B }  i^i  A )  =  { B } )
42, 3sylib 121 . . . . . 6  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( { B }  i^i  A )  =  { B } )
54imaeq2d 4876 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " ( { B }  i^i  A
) )  =  ( R " { B } ) )
65ineq1d 3271 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( ( R "
( { B }  i^i  A ) )  i^i 
A )  =  ( ( R " { B } )  i^i  A
) )
7 imass2 4910 . . . . . . 7  |-  ( { B }  C_  A  ->  ( R " { B } )  C_  ( R " A ) )
82, 7syl 14 . . . . . 6  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  C_  ( R " A ) )
9 simpl 108 . . . . . 6  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " A
)  C_  A )
108, 9sstrd 3102 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  C_  A
)
11 df-ss 3079 . . . . 5  |-  ( ( R " { B } )  C_  A  <->  ( ( R " { B } )  i^i  A
)  =  ( R
" { B }
) )
1210, 11sylib 121 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( ( R " { B } )  i^i 
A )  =  ( R " { B } ) )
136, 12eqtr2d 2171 . . 3  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  =  ( ( R " ( { B }  i^i  A
) )  i^i  A
) )
14 imainrect 4979 . . 3  |-  ( ( R  i^i  ( A  X.  A ) )
" { B }
)  =  ( ( R " ( { B }  i^i  A
) )  i^i  A
)
1513, 14syl6eqr 2188 . 2  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  =  ( ( R  i^i  ( A  X.  A ) )
" { B }
) )
16 df-ec 6424 . 2  |-  [ B ] R  =  ( R " { B }
)
17 df-ec 6424 . 2  |-  [ B ] ( R  i^i  ( A  X.  A
) )  =  ( ( R  i^i  ( A  X.  A ) )
" { B }
)
1815, 16, 173eqtr4g 2195 1  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  [ B ] R  =  [ B ] ( R  i^i  ( A  X.  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    i^i cin 3065    C_ wss 3066   {csn 3522    X. cxp 4532   "cima 4537   [cec 6420
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-ec 6424
This theorem is referenced by:  qsinxp  6498  nqnq0pi  7239
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