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Theorem kqdisj 19965
Description: A version of imain 5662 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 5497 . . . . 5  |-  ( F
" dom  ( F  |`  ( A  \  U
) ) )  =  ( F " ( A  \  U ) )
2 dmres 5292 . . . . . . 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 19958 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
54adantr 465 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  F  Fn  X )
6 fndm 5678 . . . . . . . . 9  |-  ( F  Fn  X  ->  dom  F  =  X )
75, 6syl 16 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  F  =  X )
87ineq2d 3700 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( A  \  U
)  i^i  dom  F )  =  ( ( A 
\  U )  i^i 
X ) )
92, 8syl5eq 2520 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  ( F  |`  ( A 
\  U ) )  =  ( ( A 
\  U )  i^i 
X ) )
109imaeq2d 5335 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " dom  ( F  |`  ( A  \  U
) ) )  =  ( F " (
( A  \  U
)  i^i  X )
) )
111, 10syl5eqr 2522 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( A  \  U ) )  =  ( F " (
( A  \  U
)  i^i  X )
) )
12 indif1 3742 . . . . . 6  |-  ( ( A  \  U )  i^i  X )  =  ( ( A  i^i  X )  \  U )
13 inss2 3719 . . . . . . 7  |-  ( A  i^i  X )  C_  X
14 ssdif 3639 . . . . . . 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 3534 . . . . 5  |-  ( ( A  \  U )  i^i  X )  C_  ( X  \  U )
17 imass2 5370 . . . . 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 3538 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( A  \  U ) )  C_  ( F " ( X 
\  U ) ) )
20 sslin 3724 . . 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 3627 . . . . . . 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 3626 . . . . . . . 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 19963 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  w  e.  X )  ->  (
w  e.  U  <->  ( F `  w )  e.  ( F " U ) ) )
2924, 25, 27, 28syl3anc 1228 . . . . . 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 2878 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  A. w  e.  ( X  \  U
)  -.  ( F `
 w )  e.  ( F " U
) )
32 difss 3631 . . . . 5  |-  ( X 
\  U )  C_  X
33 eleq1 2539 . . . . . . 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 6138 . . . . 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 3868 . . 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 3817 . 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 1379    e. wcel 1767   A.wral 2814   {crab 2818    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785    |-> cmpt 4505   dom cdm 4999    |` cres 5001   "cima 5002    Fn wfn 5581   ` cfv 5586  TopOnctopon 19159
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-topon 19166
This theorem is referenced by:  kqcldsat  19966  regr1lem  19972
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