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Theorem kqdisj 20746
Description: A version of imain 5677 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 5346 . . . . 5  |-  ( F
" dom  ( F  |`  ( A  \  U
) ) )  =  ( F " ( A  \  U ) )
2 dmres 5144 . . . . . . 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 20739 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
54adantr 466 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  F  Fn  X )
6 fndm 5693 . . . . . . . . 9  |-  ( F  Fn  X  ->  dom  F  =  X )
75, 6syl 17 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  F  =  X )
87ineq2d 3664 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( A  \  U
)  i^i  dom  F )  =  ( ( A 
\  U )  i^i 
X ) )
92, 8syl5eq 2475 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  ( F  |`  ( A 
\  U ) )  =  ( ( A 
\  U )  i^i 
X ) )
109imaeq2d 5187 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " dom  ( F  |`  ( A  \  U
) ) )  =  ( F " (
( A  \  U
)  i^i  X )
) )
111, 10syl5eqr 2477 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( A  \  U ) )  =  ( F " (
( A  \  U
)  i^i  X )
) )
12 indif1 3717 . . . . . 6  |-  ( ( A  \  U )  i^i  X )  =  ( ( A  i^i  X )  \  U )
13 inss2 3683 . . . . . . 7  |-  ( A  i^i  X )  C_  X
14 ssdif 3600 . . . . . . 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 3494 . . . . 5  |-  ( ( A  \  U )  i^i  X )  C_  ( X  \  U )
17 imass2 5223 . . . . 5  |-  ( ( ( A  \  U
)  i^i  X )  C_  ( X  \  U
)  ->  ( F " ( ( A  \  U )  i^i  X
) )  C_  ( F " ( X  \  U ) ) )
1816, 17mp1i 13 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( ( A 
\  U )  i^i 
X ) )  C_  ( F " ( X 
\  U ) ) )
1911, 18eqsstrd 3498 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( A  \  U ) )  C_  ( F " ( X 
\  U ) ) )
20 sslin 3688 . . 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 17 . 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 3588 . . . . . . 7  |-  ( w  e.  ( X  \  U )  ->  -.  w  e.  U )
2322adantl 467 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  -.  w  e.  U )
24 simpll 758 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  J  e.  (TopOn `  X )
)
25 simplr 760 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  U  e.  J )
26 eldifi 3587 . . . . . . . 8  |-  ( w  e.  ( X  \  U )  ->  w  e.  X )
2726adantl 467 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  w  e.  X )
283kqfvima 20744 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  w  e.  X )  ->  (
w  e.  U  <->  ( F `  w )  e.  ( F " U ) ) )
2924, 25, 27, 28syl3anc 1264 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  (
w  e.  U  <->  ( F `  w )  e.  ( F " U ) ) )
3023, 29mtbid 301 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  -.  ( F `  w )  e.  ( F " U ) )
3130ralrimiva 2836 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  A. w  e.  ( X  \  U
)  -.  ( F `
 w )  e.  ( F " U
) )
32 difss 3592 . . . . 5  |-  ( X 
\  U )  C_  X
33 eleq1 2495 . . . . . . 7  |-  ( z  =  ( F `  w )  ->  (
z  e.  ( F
" U )  <->  ( F `  w )  e.  ( F " U ) ) )
3433notbid 295 . . . . . 6  |-  ( z  =  ( F `  w )  ->  ( -.  z  e.  ( F " U )  <->  -.  ( F `  w )  e.  ( F " U
) ) )
3534ralima 6161 . . . . 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 666 . . . 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 235 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  A. z  e.  ( F " ( X  \  U ) )  -.  z  e.  ( F " U ) )
38 disjr 3836 . . 3  |-  ( ( ( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) 
<-> 
A. z  e.  ( F " ( X 
\  U ) )  -.  z  e.  ( F " U ) )
3937, 38sylibr 215 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) )
40 sseq0 3796 . 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 665 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   {crab 2775    \ cdif 3433    i^i cin 3435    C_ wss 3436   (/)c0 3761    |-> cmpt 4482   dom cdm 4853    |` cres 4855   "cima 4856    Fn wfn 5596   ` cfv 5601  TopOnctopon 19917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-topon 19922
This theorem is referenced by:  kqcldsat  20747  regr1lem  20753
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