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Theorem ecinxp 6568
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 3713 . . . . . . 7  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  { B }  C_  A )
3 df-ss 3125 . . . . . . 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 4941 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " ( { B }  i^i  A
) )  =  ( R " { B } ) )
65ineq1d 3318 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( ( R "
( { B }  i^i  A ) )  i^i 
A )  =  ( ( R " { B } )  i^i  A
) )
7 imass2 4975 . . . . . . 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 3148 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  C_  A
)
11 df-ss 3125 . . . . 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 2198 . . 3  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  =  ( ( R " ( { B }  i^i  A
) )  i^i  A
) )
14 imainrect 5044 . . 3  |-  ( ( R  i^i  ( A  X.  A ) )
" { B }
)  =  ( ( R " ( { B }  i^i  A
) )  i^i  A
)
1513, 14eqtr4di 2215 . 2  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  =  ( ( R  i^i  ( A  X.  A ) )
" { B }
) )
16 df-ec 6495 . 2  |-  [ B ] R  =  ( R " { B }
)
17 df-ec 6495 . 2  |-  [ B ] ( R  i^i  ( A  X.  A
) )  =  ( ( R  i^i  ( A  X.  A ) )
" { B }
)
1815, 16, 173eqtr4g 2222 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 1342    e. wcel 2135    i^i cin 3111    C_ wss 3112   {csn 3571    X. cxp 4597   "cima 4602   [cec 6491
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-xp 4605  df-rel 4606  df-cnv 4607  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-ec 6495
This theorem is referenced by:  qsinxp  6569  nqnq0pi  7371
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