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Theorem xkotf 19163
Description: Functionality of function  T. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
xkoval.x  |-  X  = 
U. R
xkoval.k  |-  K  =  { x  e.  ~P X  |  ( Rt  x
)  e.  Comp }
xkoval.t  |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )
Assertion
Ref Expression
xkotf  |-  T :
( K  X.  S
) --> ~P ( R  Cn  S )
Distinct variable groups:    v, k, K    f, k, v, x, R    S, f, k, v, x    k, X, x
Allowed substitution hints:    T( x, v, f, k)    K( x, f)    X( v, f)

Proof of Theorem xkotf
StepHypRef Expression
1 ssrab2 3442 . . . 4  |-  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v }  C_  ( R  Cn  S
)
2 ovex 6121 . . . . 5  |-  ( R  Cn  S )  e. 
_V
32elpw2 4461 . . . 4  |-  ( { f  e.  ( R  Cn  S )  |  ( f " k
)  C_  v }  e.  ~P ( R  Cn  S )  <->  { f  e.  ( R  Cn  S
)  |  ( f
" k )  C_  v }  C_  ( R  Cn  S ) )
41, 3mpbir 209 . . 3  |-  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v }  e.  ~P ( R  Cn  S
)
54rgen2w 2789 . 2  |-  A. k  e.  K  A. v  e.  S  { f  e.  ( R  Cn  S
)  |  ( f
" k )  C_  v }  e.  ~P ( R  Cn  S
)
6 xkoval.t . . 3  |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )
76fmpt2 6646 . 2  |-  ( A. k  e.  K  A. v  e.  S  {
f  e.  ( R  Cn  S )  |  ( f " k
)  C_  v }  e.  ~P ( R  Cn  S )  <->  T :
( K  X.  S
) --> ~P ( R  Cn  S ) )
85, 7mpbi 208 1  |-  T :
( K  X.  S
) --> ~P ( R  Cn  S )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   A.wral 2720   {crab 2724    C_ wss 3333   ~Pcpw 3865   U.cuni 4096    X. cxp 4843   "cima 4848   -->wf 5419  (class class class)co 6096    e. cmpt2 6098   ↾t crest 14364    Cn ccn 18833   Compccmp 18994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583
This theorem is referenced by:  xkoopn  19167  xkouni  19177  xkoccn  19197  xkoco1cn  19235  xkoco2cn  19236  xkococn  19238  xkoinjcn  19265
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