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

Theorem resindir 4656
Description: Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
resindir  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )

Proof of Theorem resindir
StepHypRef Expression
1 inindir 3183 . 2  |-  ( ( A  i^i  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  ( C  X.  _V ) )  i^i  ( B  i^i  ( C  X.  _V )
) )
2 df-res 4385 . 2  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  i^i  B
)  i^i  ( C  X.  _V ) )
3 df-res 4385 . . 3  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
4 df-res 4385 . . 3  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
53, 4ineq12i 3164 . 2  |-  ( ( A  |`  C )  i^i  ( B  |`  C ) )  =  ( ( A  i^i  ( C  X.  _V ) )  i^i  ( B  i^i  ( C  X.  _V )
) )
61, 2, 53eqtr4i 2086 1  |-  ( ( A  i^i  B )  |`  C )  =  ( ( A  |`  C )  i^i  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1259   _Vcvv 2574    i^i cin 2944    X. cxp 4371    |` cres 4375
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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-res 4385
This theorem is referenced by:  inimass  4768  fnreseql  5305
  Copyright terms: Public domain W3C validator