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Theorem xkobval 20600
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 6421 . 2  |-  ran  T  =  { s  |  E. k  e.  K  E. v  e.  S  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } }
3 oveq2 6314 . . . . . 6  |-  ( x  =  k  ->  ( Rt  x )  =  ( Rt  k ) )
43eleq1d 2491 . . . . 5  |-  ( x  =  k  ->  (
( Rt  x )  e.  Comp  <->  ( Rt  k )  e.  Comp ) )
54rexrab 3234 . . . 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 3027 . . . 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 2980 . . . . 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 2924 . . . 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 280 . . 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 2551 . 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 2451 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 370    = wceq 1437    e. wcel 1872   {cab 2407   E.wrex 2772   {crab 2775    C_ wss 3436   ~Pcpw 3981   U.cuni 4219   ran crn 4854   "cima 4856  (class class class)co 6306    |-> cmpt2 6308   ↾t crest 15319    Cn ccn 20239   Compccmp 20400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-cnv 4861  df-dm 4863  df-rn 4864  df-iota 5565  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311
This theorem is referenced by:  xkoccn  20633  xkoco1cn  20671  xkoco2cn  20672  xkoinjcn  20701
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