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Theorem kqsat 20100
Description: Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 20086). (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
Assertion
Ref Expression
kqsat  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  =  U )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    U( x, y)    F( x, y)

Proof of Theorem kqsat
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . . 7  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqffn 20094 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
3 elpreima 6008 . . . . . 6  |-  ( F  Fn  X  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
42, 3syl 16 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( z  e.  ( `' F "
( F " U
) )  <->  ( z  e.  X  /\  ( F `  z )  e.  ( F " U
) ) ) )
54adantr 465 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
61kqfvima 20099 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  z  e.  X )  ->  (
z  e.  U  <->  ( F `  z )  e.  ( F " U ) ) )
763expa 1196 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  z  e.  X )  ->  (
z  e.  U  <->  ( F `  z )  e.  ( F " U ) ) )
87biimprd 223 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  z  e.  X )  ->  (
( F `  z
)  e.  ( F
" U )  -> 
z  e.  U ) )
98expimpd 603 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) )  ->  z  e.  U
) )
105, 9sylbid 215 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
z  e.  ( `' F " ( F
" U ) )  ->  z  e.  U
) )
1110ssrdv 3515 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) ) 
C_  U )
12 toponss 19299 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  X )
13 fndm 5686 . . . . . . 7  |-  ( F  Fn  X  ->  dom  F  =  X )
142, 13syl 16 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  =  X )
1514adantr 465 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  F  =  X )
1612, 15sseqtr4d 3546 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_ 
dom  F )
17 dfss1 3708 . . . 4  |-  ( U 
C_  dom  F  <->  ( dom  F  i^i  U )  =  U )
1816, 17sylib 196 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( dom  F  i^i  U )  =  U )
19 dminss 5426 . . 3  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
2018, 19syl6eqssr 3560 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  ( `' F "
( F " U
) ) )
2111, 20eqssd 3526 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821    i^i cin 3480    C_ wss 3481    |-> cmpt 4511   `'ccnv 5004   dom cdm 5005   "cima 5008    Fn wfn 5589   ` cfv 5594  TopOnctopon 19264
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-topon 19271
This theorem is referenced by:  kqopn  20103  kqreglem2  20111  kqnrmlem2  20113
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