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Theorem selsubf3 25991
Description: A way of selecting a subset of functions so that their values belong to  B. (Contributed by FL, 14-Jan-2014.)
Hypotheses
Ref Expression
selsubf3.1  |-  A  e. 
_V
selsubf3.2  |-  C  e. 
_V
Assertion
Ref Expression
selsubf3  |-  ( ( A  ^m  C )  i^i  ~P ( _V 
X.  B ) )  =  ( ( A  i^i  B )  ^m  C )

Proof of Theorem selsubf3
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 in32 3381 . . 3  |-  ( ( ~P ( C  X.  A )  i^i  {
f  |  f  Fn  C } )  i^i 
~P ( _V  X.  B ) )  =  ( ( ~P ( C  X.  A )  i^i 
~P ( _V  X.  B ) )  i^i 
{ f  |  f  Fn  C } )
2 pwin 4297 . . . . 5  |-  ~P (
( C  X.  A
)  i^i  ( _V  X.  B ) )  =  ( ~P ( C  X.  A )  i^i 
~P ( _V  X.  B ) )
3 inxp 4818 . . . . . . 7  |-  ( ( C  X.  A )  i^i  ( _V  X.  B ) )  =  ( ( C  i^i  _V )  X.  ( A  i^i  B ) )
4 inv1 3481 . . . . . . . 8  |-  ( C  i^i  _V )  =  C
54xpeq1i 4709 . . . . . . 7  |-  ( ( C  i^i  _V )  X.  ( A  i^i  B
) )  =  ( C  X.  ( A  i^i  B ) )
63, 5eqtri 2303 . . . . . 6  |-  ( ( C  X.  A )  i^i  ( _V  X.  B ) )  =  ( C  X.  ( A  i^i  B ) )
76pweqi 3629 . . . . 5  |-  ~P (
( C  X.  A
)  i^i  ( _V  X.  B ) )  =  ~P ( C  X.  ( A  i^i  B ) )
82, 7eqtr3i 2305 . . . 4  |-  ( ~P ( C  X.  A
)  i^i  ~P ( _V  X.  B ) )  =  ~P ( C  X.  ( A  i^i  B ) )
98ineq1i 3366 . . 3  |-  ( ( ~P ( C  X.  A )  i^i  ~P ( _V  X.  B
) )  i^i  {
f  |  f  Fn  C } )  =  ( ~P ( C  X.  ( A  i^i  B ) )  i^i  {
f  |  f  Fn  C } )
101, 9eqtri 2303 . 2  |-  ( ( ~P ( C  X.  A )  i^i  {
f  |  f  Fn  C } )  i^i 
~P ( _V  X.  B ) )  =  ( ~P ( C  X.  ( A  i^i  B ) )  i^i  {
f  |  f  Fn  C } )
11 selsubf3.1 . . . 4  |-  A  e. 
_V
12 selsubf3.2 . . . 4  |-  C  e. 
_V
1311, 12mapval2 6797 . . 3  |-  ( A  ^m  C )  =  ( ~P ( C  X.  A )  i^i 
{ f  |  f  Fn  C } )
1413ineq1i 3366 . 2  |-  ( ( A  ^m  C )  i^i  ~P ( _V 
X.  B ) )  =  ( ( ~P ( C  X.  A
)  i^i  { f  |  f  Fn  C } )  i^i  ~P ( _V  X.  B
) )
1511inex1 4155 . . 3  |-  ( A  i^i  B )  e. 
_V
1615, 12mapval2 6797 . 2  |-  ( ( A  i^i  B )  ^m  C )  =  ( ~P ( C  X.  ( A  i^i  B ) )  i^i  {
f  |  f  Fn  C } )
1710, 14, 163eqtr4i 2313 1  |-  ( ( A  ^m  C )  i^i  ~P ( _V 
X.  B ) )  =  ( ( A  i^i  B )  ^m  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    i^i cin 3151   ~Pcpw 3625    X. cxp 4687    Fn wfn 5250  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  selsubf3g  25992  pgapspf2  26053
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774
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