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Theorem xkotf 20268
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 3521 . . . 4  |-  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v }  C_  ( R  Cn  S
)
2 ovex 6260 . . . . 5  |-  ( R  Cn  S )  e. 
_V
32elpw2 4555 . . . 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 2763 . 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 6803 . 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 1403    e. wcel 1840   A.wral 2751   {crab 2755    C_ wss 3411   ~Pcpw 3952   U.cuni 4188    X. cxp 4938   "cima 4943   -->wf 5519  (class class class)co 6232    |-> cmpt2 6234   ↾t crest 14925    Cn ccn 19908   Compccmp 20069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737
This theorem is referenced by:  xkoopn  20272  xkouni  20282  xkoccn  20302  xkoco1cn  20340  xkoco2cn  20341  xkococn  20343  xkoinjcn  20370
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