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Theorem xkotf 19821
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 3585 . . . 4  |-  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v }  C_  ( R  Cn  S
)
2 ovex 6307 . . . . 5  |-  ( R  Cn  S )  e. 
_V
32elpw2 4611 . . . 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 2826 . 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 6848 . 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 1379    e. wcel 1767   A.wral 2814   {crab 2818    C_ wss 3476   ~Pcpw 4010   U.cuni 4245    X. cxp 4997   "cima 5002   -->wf 5582  (class class class)co 6282    |-> cmpt2 6284   ↾t crest 14672    Cn ccn 19491   Compccmp 19652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782
This theorem is referenced by:  xkoopn  19825  xkouni  19835  xkoccn  19855  xkoco1cn  19893  xkoco2cn  19894  xkococn  19896  xkoinjcn  19923
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