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Theorem kqsat 19446
Description: Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 19432). (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 19440 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
3 elpreima 5935 . . . . . 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 19445 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  z  e.  X )  ->  (
z  e.  U  <->  ( F `  z )  e.  ( F " U ) ) )
763expa 1188 . . . . . 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 3473 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) ) 
C_  U )
12 toponss 18676 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  X )
13 fndm 5621 . . . . . . 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 3504 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_ 
dom  F )
17 dfss1 3666 . . . 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 5362 . . 3  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
2018, 19syl6eqssr 3518 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  ( `' F "
( F " U
) ) )
2111, 20eqssd 3484 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 1370    e. wcel 1758   {crab 2803    i^i cin 3438    C_ wss 3439    |-> cmpt 4461   `'ccnv 4950   dom cdm 4951   "cima 4954    Fn wfn 5524   ` cfv 5529  TopOnctopon 18641
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-topon 18648
This theorem is referenced by:  kqopn  19449  kqreglem2  19457  kqnrmlem2  19459
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