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Theorem xkobval 19850
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 6396 . 2  |-  ran  T  =  { s  |  E. k  e.  K  E. v  e.  S  s  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } }
3 oveq2 6292 . . . . . 6  |-  ( x  =  k  ->  ( Rt  x )  =  ( Rt  k ) )
43eleq1d 2536 . . . . 5  |-  ( x  =  k  ->  (
( Rt  x )  e.  Comp  <->  ( Rt  k )  e.  Comp ) )
54rexrab 3267 . . . 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 3063 . . . 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 3016 . . . . 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 2965 . . . 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 2601 . 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 2496 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 1379    e. wcel 1767   {cab 2452   E.wrex 2815   {crab 2818    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   ran crn 5000   "cima 5002  (class class class)co 6284    |-> cmpt2 6286   ↾t crest 14676    Cn ccn 19519   Compccmp 19680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010  df-iota 5551  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289
This theorem is referenced by:  xkoccn  19883  xkoco1cn  19921  xkoco2cn  19922  xkoinjcn  19951
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