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Theorem kqsat 17755
Description: Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17741). (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 17749 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
3 elpreima 5842 . . . . . 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 452 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
61kqfvima 17754 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  z  e.  X )  ->  (
z  e.  U  <->  ( F `  z )  e.  ( F " U ) ) )
763expa 1153 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  z  e.  X )  ->  (
z  e.  U  <->  ( F `  z )  e.  ( F " U ) ) )
87biimprd 215 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  z  e.  X )  ->  (
( F `  z
)  e.  ( F
" U )  -> 
z  e.  U ) )
98expimpd 587 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) )  ->  z  e.  U
) )
105, 9sylbid 207 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
z  e.  ( `' F " ( F
" U ) )  ->  z  e.  U
) )
1110ssrdv 3346 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) ) 
C_  U )
12 toponss 16986 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  X )
13 fndm 5536 . . . . . . 7  |-  ( F  Fn  X  ->  dom  F  =  X )
142, 13syl 16 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  =  X )
1514adantr 452 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  F  =  X )
1612, 15sseqtr4d 3377 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_ 
dom  F )
17 dfss1 3537 . . . 4  |-  ( U 
C_  dom  F  <->  ( dom  F  i^i  U )  =  U )
1816, 17sylib 189 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( dom  F  i^i  U )  =  U )
19 dminss 5278 . . 3  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
2018, 19syl6eqssr 3391 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  ( `' F "
( F " U
) ) )
2111, 20eqssd 3357 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701    i^i cin 3311    C_ wss 3312    e. cmpt 4258   `'ccnv 4869   dom cdm 4870   "cima 4873    Fn wfn 5441   ` cfv 5446  TopOnctopon 16951
This theorem is referenced by:  kqopn  17758  kqreglem2  17766  kqnrmlem2  17768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-topon 16958
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