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Theorem selsubf3g 25992
Description: A way of selecting a subset of functions so that their values belong to  B. (Contributed by FL, 14-Jan-2014.)
Assertion
Ref Expression
selsubf3g  |-  ( ( A  e.  D  /\  C  e.  E )  ->  ( ( A  ^m  C )  i^i  ~P ( _V  X.  B
) )  =  ( ( A  i^i  B
)  ^m  C )
)

Proof of Theorem selsubf3g
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  D  ->  A  e.  _V )
2 elex 2796 . 2  |-  ( C  e.  E  ->  C  e.  _V )
3 oveq1 5865 . . . . 5  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  ( A  ^m  C )  =  ( if ( A  e.  _V ,  A ,  (/) )  ^m  C
) )
43ineq1d 3369 . . . 4  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  (
( A  ^m  C
)  i^i  ~P ( _V  X.  B ) )  =  ( ( if ( A  e.  _V ,  A ,  (/) )  ^m  C )  i^i  ~P ( _V  X.  B
) ) )
5 ineq1 3363 . . . . 5  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  ( A  i^i  B )  =  ( if ( A  e.  _V ,  A ,  (/) )  i^i  B
) )
65oveq1d 5873 . . . 4  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  (
( A  i^i  B
)  ^m  C )  =  ( ( if ( A  e.  _V ,  A ,  (/) )  i^i 
B )  ^m  C
) )
74, 6eqeq12d 2297 . . 3  |-  ( A  =  if ( A  e.  _V ,  A ,  (/) )  ->  (
( ( A  ^m  C )  i^i  ~P ( _V  X.  B
) )  =  ( ( A  i^i  B
)  ^m  C )  <->  ( ( if ( A  e.  _V ,  A ,  (/) )  ^m  C
)  i^i  ~P ( _V  X.  B ) )  =  ( ( if ( A  e.  _V ,  A ,  (/) )  i^i 
B )  ^m  C
) ) )
8 oveq2 5866 . . . . 5  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  ( if ( A  e.  _V ,  A ,  (/) )  ^m  C )  =  ( if ( A  e. 
_V ,  A ,  (/) )  ^m  if ( C  e.  _V ,  C ,  (/) ) ) )
98ineq1d 3369 . . . 4  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
( if ( A  e.  _V ,  A ,  (/) )  ^m  C
)  i^i  ~P ( _V  X.  B ) )  =  ( ( if ( A  e.  _V ,  A ,  (/) )  ^m  if ( C  e.  _V ,  C ,  (/) ) )  i^i  ~P ( _V 
X.  B ) ) )
10 oveq2 5866 . . . 4  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
( if ( A  e.  _V ,  A ,  (/) )  i^i  B
)  ^m  C )  =  ( ( if ( A  e.  _V ,  A ,  (/) )  i^i 
B )  ^m  if ( C  e.  _V ,  C ,  (/) ) ) )
119, 10eqeq12d 2297 . . 3  |-  ( C  =  if ( C  e.  _V ,  C ,  (/) )  ->  (
( ( if ( A  e.  _V ,  A ,  (/) )  ^m  C )  i^i  ~P ( _V  X.  B
) )  =  ( ( if ( A  e.  _V ,  A ,  (/) )  i^i  B
)  ^m  C )  <->  ( ( if ( A  e.  _V ,  A ,  (/) )  ^m  if ( C  e.  _V ,  C ,  (/) ) )  i^i  ~P ( _V 
X.  B ) )  =  ( ( if ( A  e.  _V ,  A ,  (/) )  i^i 
B )  ^m  if ( C  e.  _V ,  C ,  (/) ) ) ) )
12 0ex 4150 . . . . 5  |-  (/)  e.  _V
1312elimel 3617 . . . 4  |-  if ( A  e.  _V ,  A ,  (/) )  e. 
_V
1412elimel 3617 . . . 4  |-  if ( C  e.  _V ,  C ,  (/) )  e. 
_V
1513, 14selsubf3 25991 . . 3  |-  ( ( if ( A  e. 
_V ,  A ,  (/) )  ^m  if ( C  e.  _V ,  C ,  (/) ) )  i^i  ~P ( _V 
X.  B ) )  =  ( ( if ( A  e.  _V ,  A ,  (/) )  i^i 
B )  ^m  if ( C  e.  _V ,  C ,  (/) ) )
167, 11, 15dedth2h 3607 . 2  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( ( A  ^m  C )  i^i  ~P ( _V  X.  B
) )  =  ( ( A  i^i  B
)  ^m  C )
)
171, 2, 16syl2an 463 1  |-  ( ( A  e.  D  /\  C  e.  E )  ->  ( ( A  ^m  C )  i^i  ~P ( _V  X.  B
) )  =  ( ( A  i^i  B
)  ^m  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   (/)c0 3455   ifcif 3565   ~Pcpw 3625    X. cxp 4687  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  indcls2  25996
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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