ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ecinxp Unicode version

Theorem ecinxp 6212
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 107 . . . . . . . 8  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  B  e.  A )
21snssd 3537 . . . . . . 7  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  { B }  C_  A )
3 df-ss 2959 . . . . . . 7  |-  ( { B }  C_  A  <->  ( { B }  i^i  A )  =  { B } )
42, 3sylib 131 . . . . . 6  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( { B }  i^i  A )  =  { B } )
54imaeq2d 4696 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " ( { B }  i^i  A
) )  =  ( R " { B } ) )
65ineq1d 3165 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( ( R "
( { B }  i^i  A ) )  i^i 
A )  =  ( ( R " { B } )  i^i  A
) )
7 imass2 4729 . . . . . . 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 106 . . . . . 6  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " A
)  C_  A )
108, 9sstrd 2983 . . . . 5  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  C_  A
)
11 df-ss 2959 . . . . 5  |-  ( ( R " { B } )  C_  A  <->  ( ( R " { B } )  i^i  A
)  =  ( R
" { B }
) )
1210, 11sylib 131 . . . 4  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( ( R " { B } )  i^i 
A )  =  ( R " { B } ) )
136, 12eqtr2d 2089 . . 3  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  =  ( ( R " ( { B }  i^i  A
) )  i^i  A
) )
14 imainrect 4794 . . 3  |-  ( ( R  i^i  ( A  X.  A ) )
" { B }
)  =  ( ( R " ( { B }  i^i  A
) )  i^i  A
)
1513, 14syl6eqr 2106 . 2  |-  ( ( ( R " A
)  C_  A  /\  B  e.  A )  ->  ( R " { B } )  =  ( ( R  i^i  ( A  X.  A ) )
" { B }
) )
16 df-ec 6139 . 2  |-  [ B ] R  =  ( R " { B }
)
17 df-ec 6139 . 2  |-  [ B ] ( R  i^i  ( A  X.  A
) )  =  ( ( R  i^i  ( A  X.  A ) )
" { B }
)
1815, 16, 173eqtr4g 2113 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 101    = wceq 1259    e. wcel 1409    i^i cin 2944    C_ wss 2945   {csn 3403    X. cxp 4371   "cima 4376   [cec 6135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-ec 6139
This theorem is referenced by:  qsinxp  6213  nqnq0pi  6594
  Copyright terms: Public domain W3C validator