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Theorem kqdisj 19436
Description: A version of imain 5601 for the topological indistinguishability map. (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
kqdisj  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( F " U
)  i^i  ( F " ( A  \  U
) ) )  =  (/) )
Distinct variable groups:    x, y, A    x, J, y    x, X, y
Allowed substitution hints:    U( x, y)    F( x, y)

Proof of Theorem kqdisj
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imadmres 5437 . . . . 5  |-  ( F
" dom  ( F  |`  ( A  \  U
) ) )  =  ( F " ( A  \  U ) )
2 dmres 5238 . . . . . . 7  |-  dom  ( F  |`  ( A  \  U ) )  =  ( ( A  \  U )  i^i  dom  F )
3 kqval.2 . . . . . . . . . . 11  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
43kqffn 19429 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
54adantr 465 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  F  Fn  X )
6 fndm 5617 . . . . . . . . 9  |-  ( F  Fn  X  ->  dom  F  =  X )
75, 6syl 16 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  F  =  X )
87ineq2d 3659 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( A  \  U
)  i^i  dom  F )  =  ( ( A 
\  U )  i^i 
X ) )
92, 8syl5eq 2507 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  ( F  |`  ( A 
\  U ) )  =  ( ( A 
\  U )  i^i 
X ) )
109imaeq2d 5276 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " dom  ( F  |`  ( A  \  U
) ) )  =  ( F " (
( A  \  U
)  i^i  X )
) )
111, 10syl5eqr 2509 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( A  \  U ) )  =  ( F " (
( A  \  U
)  i^i  X )
) )
12 indif1 3701 . . . . . 6  |-  ( ( A  \  U )  i^i  X )  =  ( ( A  i^i  X )  \  U )
13 inss2 3678 . . . . . . 7  |-  ( A  i^i  X )  C_  X
14 ssdif 3598 . . . . . . 7  |-  ( ( A  i^i  X ) 
C_  X  ->  (
( A  i^i  X
)  \  U )  C_  ( X  \  U
) )
1513, 14ax-mp 5 . . . . . 6  |-  ( ( A  i^i  X ) 
\  U )  C_  ( X  \  U )
1612, 15eqsstri 3493 . . . . 5  |-  ( ( A  \  U )  i^i  X )  C_  ( X  \  U )
17 imass2 5311 . . . . 5  |-  ( ( ( A  \  U
)  i^i  X )  C_  ( X  \  U
)  ->  ( F " ( ( A  \  U )  i^i  X
) )  C_  ( F " ( X  \  U ) ) )
1816, 17mp1i 12 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( ( A 
\  U )  i^i 
X ) )  C_  ( F " ( X 
\  U ) ) )
1911, 18eqsstrd 3497 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( A  \  U ) )  C_  ( F " ( X 
\  U ) ) )
20 sslin 3683 . . 3  |-  ( ( F " ( A 
\  U ) ) 
C_  ( F "
( X  \  U
) )  ->  (
( F " U
)  i^i  ( F " ( A  \  U
) ) )  C_  ( ( F " U )  i^i  ( F " ( X  \  U ) ) ) )
2119, 20syl 16 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( F " U
)  i^i  ( F " ( A  \  U
) ) )  C_  ( ( F " U )  i^i  ( F " ( X  \  U ) ) ) )
22 eldifn 3586 . . . . . . 7  |-  ( w  e.  ( X  \  U )  ->  -.  w  e.  U )
2322adantl 466 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  -.  w  e.  U )
24 simpll 753 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  J  e.  (TopOn `  X )
)
25 simplr 754 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  U  e.  J )
26 eldifi 3585 . . . . . . . 8  |-  ( w  e.  ( X  \  U )  ->  w  e.  X )
2726adantl 466 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  w  e.  X )
283kqfvima 19434 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  w  e.  X )  ->  (
w  e.  U  <->  ( F `  w )  e.  ( F " U ) ) )
2924, 25, 27, 28syl3anc 1219 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  (
w  e.  U  <->  ( F `  w )  e.  ( F " U ) ) )
3023, 29mtbid 300 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  -.  ( F `  w )  e.  ( F " U ) )
3130ralrimiva 2829 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  A. w  e.  ( X  \  U
)  -.  ( F `
 w )  e.  ( F " U
) )
32 difss 3590 . . . . 5  |-  ( X 
\  U )  C_  X
33 eleq1 2526 . . . . . . 7  |-  ( z  =  ( F `  w )  ->  (
z  e.  ( F
" U )  <->  ( F `  w )  e.  ( F " U ) ) )
3433notbid 294 . . . . . 6  |-  ( z  =  ( F `  w )  ->  ( -.  z  e.  ( F " U )  <->  -.  ( F `  w )  e.  ( F " U
) ) )
3534ralima 6065 . . . . 5  |-  ( ( F  Fn  X  /\  ( X  \  U ) 
C_  X )  -> 
( A. z  e.  ( F " ( X  \  U ) )  -.  z  e.  ( F " U )  <->  A. w  e.  ( X  \  U )  -.  ( F `  w
)  e.  ( F
" U ) ) )
365, 32, 35sylancl 662 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( A. z  e.  ( F " ( X  \  U ) )  -.  z  e.  ( F
" U )  <->  A. w  e.  ( X  \  U
)  -.  ( F `
 w )  e.  ( F " U
) ) )
3731, 36mpbird 232 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  A. z  e.  ( F " ( X  \  U ) )  -.  z  e.  ( F " U ) )
38 disjr 3827 . . 3  |-  ( ( ( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) 
<-> 
A. z  e.  ( F " ( X 
\  U ) )  -.  z  e.  ( F " U ) )
3937, 38sylibr 212 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) )
40 sseq0 3776 . 2  |-  ( ( ( ( F " U )  i^i  ( F " ( A  \  U ) ) ) 
C_  ( ( F
" U )  i^i  ( F " ( X  \  U ) ) )  /\  ( ( F " U )  i^i  ( F "
( X  \  U
) ) )  =  (/) )  ->  ( ( F " U )  i^i  ( F "
( A  \  U
) ) )  =  (/) )
4121, 39, 40syl2anc 661 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( F " U
)  i^i  ( F " ( A  \  U
) ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2798   {crab 2802    \ cdif 3432    i^i cin 3434    C_ wss 3435   (/)c0 3744    |-> cmpt 4457   dom cdm 4947    |` cres 4949   "cima 4950    Fn wfn 5520   ` cfv 5525  TopOnctopon 18630
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 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-topon 18637
This theorem is referenced by:  kqcldsat  19437  regr1lem  19443
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