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Theorem xkotf 19117
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 3434 . . . 4  |-  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v }  C_  ( R  Cn  S
)
2 ovex 6115 . . . . 5  |-  ( R  Cn  S )  e. 
_V
32elpw2 4453 . . . 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 2782 . 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 6640 . 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 1364    e. wcel 1761   A.wral 2713   {crab 2717    C_ wss 3325   ~Pcpw 3857   U.cuni 4088    X. cxp 4834   "cima 4839   -->wf 5411  (class class class)co 6090    e. cmpt2 6092   ↾t crest 14355    Cn ccn 18787   Compccmp 18948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577
This theorem is referenced by:  xkoopn  19121  xkouni  19131  xkoccn  19151  xkoco1cn  19189  xkoco2cn  19190  xkococn  19192  xkoinjcn  19219
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