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Theorem xkobval 19290
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 6309 . 2  |-  ran  T  =  { s  |  E. k  e.  K  E. v  e.  S  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } }
3 oveq2 6207 . . . . . 6  |-  ( x  =  k  ->  ( Rt  x )  =  ( Rt  k ) )
43eleq1d 2523 . . . . 5  |-  ( x  =  k  ->  (
( Rt  x )  e.  Comp  <->  ( Rt  k )  e.  Comp ) )
54rexrab 3228 . . . 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 3026 . . . 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 2979 . . . . 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 2858 . . . 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 277 . . 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 2588 . 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 2483 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 setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439   E.wrex 2799   {crab 2802    C_ wss 3435   ~Pcpw 3967   U.cuni 4198   ran crn 4948   "cima 4950  (class class class)co 6199    |-> cmpt2 6201   ↾t crest 14477    Cn ccn 18959   Compccmp 19120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-cnv 4955  df-dm 4957  df-rn 4958  df-iota 5488  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204
This theorem is referenced by:  xkoccn  19323  xkoco1cn  19361  xkoco2cn  19362  xkoinjcn  19391
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