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Theorem ecinxp 6750
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 447 . . . . . . . 8  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  B  e.  A )
21snssd 3776 . . . . . . 7  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  { B }  C_  A )
3 df-ss 3179 . . . . . . 7  |-  ( { B }  C_  A  <->  ( { B }  i^i  A )  =  { B } )
42, 3sylib 188 . . . . . 6  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( { B }  i^i  A )  =  { B } )
54imaeq2d 5028 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " ( { B }  i^i  A
) )  =  ( R " { B } ) )
65ineq1d 3382 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( ( R "
( { B }  i^i  A ) )  i^i 
A )  =  ( ( R " { B } )  i^i  A
) )
7 imass2 5065 . . . . . . 7  |-  ( { B }  C_  A  ->  ( R " { B } )  C_  ( R " A ) )
82, 7syl 15 . . . . . 6  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  C_  ( R " A ) )
9 simpl 443 . . . . . 6  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " A
)  C_  A )
108, 9sstrd 3202 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  C_  A
)
11 df-ss 3179 . . . . 5  |-  ( ( R " { B } )  C_  A  <->  ( ( R " { B } )  i^i  A
)  =  ( R
" { B }
) )
1210, 11sylib 188 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( ( R " { B } )  i^i 
A )  =  ( R " { B } ) )
136, 12eqtr2d 2329 . . 3  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  =  ( ( R " ( { B }  i^i  A
) )  i^i  A
) )
14 imainrect 5135 . . 3  |-  ( ( R  i^i  ( A  X.  A ) )
" { B }
)  =  ( ( R " ( { B }  i^i  A
) )  i^i  A
)
1513, 14syl6eqr 2346 . 2  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  =  ( ( R  i^i  ( A  X.  A ) )
" { B }
) )
16 df-ec 6678 . 2  |-  [ B ] R  =  ( R " { B }
)
17 df-ec 6678 . 2  |-  [ B ] ( R  i^i  ( A  X.  A
) )  =  ( ( R  i^i  ( A  X.  A ) )
" { B }
)
1815, 16, 173eqtr4g 2353 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 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   {csn 3653    X. cxp 4703   "cima 4708   [cec 6674
This theorem is referenced by:  qsinxp  6751  divsin  13462  pi1addval  18562  pi1grplem  18563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-ec 6678
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