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Theorem xkobval 17571
Description: Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-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
xkobval  |-  ran  T  =  { s  |  E. k  e.  ~P  X E. v  e.  S  ( ( Rt  k )  e.  Comp  /\  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) }
Distinct variable groups:    k, s,
v, K    f, k,
s, v, x, R    S, f, k, s, v, x    T, s    k, X, x
Allowed substitution hints:    T( x, v, f, k)    K( x, f)    X( v, f, s)

Proof of Theorem xkobval
StepHypRef Expression
1 xkoval.t . . 3  |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )
21rnmpt2 6139 . 2  |-  ran  T  =  { s  |  E. k  e.  K  E. v  e.  S  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } }
3 oveq2 6048 . . . . . 6  |-  ( x  =  k  ->  ( Rt  x )  =  ( Rt  k ) )
43eleq1d 2470 . . . . 5  |-  ( x  =  k  ->  (
( Rt  x )  e.  Comp  <->  ( Rt  k )  e.  Comp ) )
54rexrab 3058 . . . 4  |-  ( E. k  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } E. v  e.  S  s  =  { f  e.  ( R  Cn  S
)  |  ( f
" k )  C_  v }  <->  E. k  e.  ~P  X ( ( Rt  k )  e.  Comp  /\  E. v  e.  S  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) )
6 xkoval.k . . . . 5  |-  K  =  { x  e.  ~P X  |  ( Rt  x
)  e.  Comp }
76rexeqi 2869 . . . 4  |-  ( E. k  e.  K  E. v  e.  S  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } 
<->  E. k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } E. v  e.  S  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )
8 r19.42v 2822 . . . . 5  |-  ( E. v  e.  S  ( ( Rt  k )  e. 
Comp  /\  s  =  {
f  e.  ( R  Cn  S )  |  ( f " k
)  C_  v }
)  <->  ( ( Rt  k )  e.  Comp  /\  E. v  e.  S  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) )
98rexbii 2691 . . . 4  |-  ( E. k  e.  ~P  X E. v  e.  S  ( ( Rt  k )  e.  Comp  /\  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )  <->  E. k  e.  ~P  X ( ( Rt  k )  e.  Comp  /\ 
E. v  e.  S  s  =  { f  e.  ( R  Cn  S
)  |  ( f
" k )  C_  v } ) )
105, 7, 93bitr4i 269 . . 3  |-  ( E. k  e.  K  E. v  e.  S  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } 
<->  E. k  e.  ~P  X E. v  e.  S  ( ( Rt  k )  e.  Comp  /\  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) )
1110abbii 2516 . 2  |-  { s  |  E. k  e.  K  E. v  e.  S  s  =  {
f  e.  ( R  Cn  S )  |  ( f " k
)  C_  v } }  =  { s  |  E. k  e.  ~P  X E. v  e.  S  ( ( Rt  k )  e.  Comp  /\  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) }
122, 11eqtri 2424 1  |-  ran  T  =  { s  |  E. k  e.  ~P  X E. v  e.  S  ( ( Rt  k )  e.  Comp  /\  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667   {crab 2670    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   ran crn 4838   "cima 4840  (class class class)co 6040    e. cmpt2 6042   ↾t crest 13603    Cn ccn 17242   Compccmp 17403
This theorem is referenced by:  xkoccn  17604  xkoco1cn  17642  xkoco2cn  17643  xkoinjcn  17672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-cnv 4845  df-dm 4847  df-rn 4848  df-iota 5377  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045
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