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Theorem kqsat 17422
Description: Any open set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17408). (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 17416 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
3 elpreima 5645 . . . . . 6  |-  ( F  Fn  X  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
42, 3syl 15 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( z  e.  ( `' F "
( F " U
) )  <->  ( z  e.  X  /\  ( F `  z )  e.  ( F " U
) ) ) )
54adantr 451 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
61kqfvima 17421 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  z  e.  X )  ->  (
z  e.  U  <->  ( F `  z )  e.  ( F " U ) ) )
763expa 1151 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  z  e.  X )  ->  (
z  e.  U  <->  ( F `  z )  e.  ( F " U ) ) )
87biimprd 214 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  z  e.  X )  ->  (
( F `  z
)  e.  ( F
" U )  -> 
z  e.  U ) )
98expimpd 586 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) )  ->  z  e.  U
) )
105, 9sylbid 206 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
z  e.  ( `' F " ( F
" U ) )  ->  z  e.  U
) )
1110ssrdv 3185 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) ) 
C_  U )
12 dminss 5095 . . 3  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
13 toponss 16667 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  X )
14 fndm 5343 . . . . . . . 8  |-  ( F  Fn  X  ->  dom  F  =  X )
152, 14syl 15 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  =  X )
1615adantr 451 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  F  =  X )
1713, 16sseqtr4d 3215 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_ 
dom  F )
18 dfss1 3373 . . . . 5  |-  ( U 
C_  dom  F  <->  ( dom  F  i^i  U )  =  U )
1917, 18sylib 188 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( dom  F  i^i  U )  =  U )
2019sseq1d 3205 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( dom  F  i^i  U )  C_  ( `' F " ( F " U ) )  <->  U  C_  ( `' F " ( F
" U ) ) ) )
2112, 20mpbii 202 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  C_  ( `' F "
( F " U
) ) )
2211, 21eqssd 3196 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    i^i cin 3151    C_ wss 3152    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   "cima 4692    Fn wfn 5250   ` cfv 5255  TopOnctopon 16632
This theorem is referenced by:  kqopn  17425  kqreglem2  17433  kqnrmlem2  17435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-topon 16639
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