ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resiun1 Unicode version

Theorem resiun1 4808
Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
resiun1  |-  ( U_ x  e.  A  B  |`  C )  =  U_ x  e.  A  ( B  |`  C )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem resiun1
StepHypRef Expression
1 iunin2 3846 . 2  |-  U_ x  e.  A  ( ( C  X.  _V )  i^i 
B )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
2 df-res 4521 . . . . 5  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
3 incom 3238 . . . . 5  |-  ( B  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  B )
42, 3eqtri 2138 . . . 4  |-  ( B  |`  C )  =  ( ( C  X.  _V )  i^i  B )
54a1i 9 . . 3  |-  ( x  e.  A  ->  ( B  |`  C )  =  ( ( C  X.  _V )  i^i  B ) )
65iuneq2i 3801 . 2  |-  U_ x  e.  A  ( B  |`  C )  =  U_ x  e.  A  (
( C  X.  _V )  i^i  B )
7 df-res 4521 . . 3  |-  ( U_ x  e.  A  B  |`  C )  =  (
U_ x  e.  A  B  i^i  ( C  X.  _V ) )
8 incom 3238 . . 3  |-  ( U_ x  e.  A  B  i^i  ( C  X.  _V ) )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
97, 8eqtri 2138 . 2  |-  ( U_ x  e.  A  B  |`  C )  =  ( ( C  X.  _V )  i^i  U_ x  e.  A  B )
101, 6, 93eqtr4ri 2149 1  |-  ( U_ x  e.  A  B  |`  C )  =  U_ x  e.  A  ( B  |`  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465   _Vcvv 2660    i^i cin 3040   U_ciun 3783    X. cxp 4507    |` cres 4511
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-iun 3785  df-res 4521
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator