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Theorem resiundiOLD 4745
Description: Obsolete proof of resiun2 4975 as of 5-Apr-2016. Distributive law for cross product over restriction. (Contributed by Mario Carneiro, 11-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
resiundiOLD  |-  ( F  |`  U_ x  e.  A  B )  =  U_ x  e.  A  ( F  |`  B )
Distinct variable group:    x, F
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem resiundiOLD
StepHypRef Expression
1 xpiundir 4744 . . . 4  |-  ( U_ x  e.  A  B  X.  _V )  =  U_ x  e.  A  ( B  X.  _V )
21ineq2i 3367 . . 3  |-  ( F  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  ( F  i^i  U_ x  e.  A  ( B  X.  _V ) )
3 iunin2 3966 . . 3  |-  U_ x  e.  A  ( F  i^i  ( B  X.  _V ) )  =  ( F  i^i  U_ x  e.  A  ( B  X.  _V ) )
42, 3eqtr4i 2306 . 2  |-  ( F  i^i  ( U_ x  e.  A  B  X.  _V ) )  =  U_ x  e.  A  ( F  i^i  ( B  X.  _V ) )
5 df-res 4701 . 2  |-  ( F  |`  U_ x  e.  A  B )  =  ( F  i^i  ( U_ x  e.  A  B  X.  _V ) )
6 df-res 4701 . . . 4  |-  ( F  |`  B )  =  ( F  i^i  ( B  X.  _V ) )
76a1i 10 . . 3  |-  ( x  e.  A  ->  ( F  |`  B )  =  ( F  i^i  ( B  X.  _V ) ) )
87iuneq2i 3923 . 2  |-  U_ x  e.  A  ( F  |`  B )  =  U_ x  e.  A  ( F  i^i  ( B  X.  _V ) )
94, 5, 83eqtr4i 2313 1  |-  ( F  |`  U_ x  e.  A  B )  =  U_ x  e.  A  ( F  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   U_ciun 3905    X. cxp 4687    |` cres 4691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iun 3907  df-opab 4078  df-xp 4695  df-res 4701
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