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Theorem kqdisj 17439
Description: A version of imain 5344 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 5181 . . . . 5  |-  ( F
" dom  ( F  |`  ( A  \  U
) ) )  =  ( F " ( A  \  U ) )
2 dmres 4992 . . . . . . 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 17432 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
54adantr 451 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  F  Fn  X )
6 fndm 5359 . . . . . . . . 9  |-  ( F  Fn  X  ->  dom  F  =  X )
75, 6syl 15 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  F  =  X )
87ineq2d 3383 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( A  \  U
)  i^i  dom  F )  =  ( ( A 
\  U )  i^i 
X ) )
92, 8syl5eq 2340 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  dom  ( F  |`  ( A 
\  U ) )  =  ( ( A 
\  U )  i^i 
X ) )
109imaeq2d 5028 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " dom  ( F  |`  ( A  \  U
) ) )  =  ( F " (
( A  \  U
)  i^i  X )
) )
111, 10syl5eqr 2342 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( A  \  U ) )  =  ( F " (
( A  \  U
)  i^i  X )
) )
12 indif1 3426 . . . . . 6  |-  ( ( A  \  U )  i^i  X )  =  ( ( A  i^i  X )  \  U )
13 inss2 3403 . . . . . . 7  |-  ( A  i^i  X )  C_  X
14 ssdif 3324 . . . . . . 7  |-  ( ( A  i^i  X ) 
C_  X  ->  (
( A  i^i  X
)  \  U )  C_  ( X  \  U
) )
1513, 14ax-mp 8 . . . . . 6  |-  ( ( A  i^i  X ) 
\  U )  C_  ( X  \  U )
1612, 15eqsstri 3221 . . . . 5  |-  ( ( A  \  U )  i^i  X )  C_  ( X  \  U )
17 imass2 5065 . . . . 5  |-  ( ( ( A  \  U
)  i^i  X )  C_  ( X  \  U
)  ->  ( F " ( ( A  \  U )  i^i  X
) )  C_  ( F " ( X  \  U ) ) )
1816, 17mp1i 11 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( ( A 
\  U )  i^i 
X ) )  C_  ( F " ( X 
\  U ) ) )
1911, 18eqsstrd 3225 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " ( A  \  U ) )  C_  ( F " ( X 
\  U ) ) )
20 sslin 3408 . . 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 15 . 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 3312 . . . . . . 7  |-  ( w  e.  ( X  \  U )  ->  -.  w  e.  U )
2322adantl 452 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  -.  w  e.  U )
24 simpll 730 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  J  e.  (TopOn `  X )
)
25 simplr 731 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  U  e.  J )
26 eldifi 3311 . . . . . . . 8  |-  ( w  e.  ( X  \  U )  ->  w  e.  X )
2726adantl 452 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  w  e.  X )
283kqfvima 17437 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J  /\  w  e.  X )  ->  (
w  e.  U  <->  ( F `  w )  e.  ( F " U ) ) )
2924, 25, 27, 28syl3anc 1182 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  (
w  e.  U  <->  ( F `  w )  e.  ( F " U ) ) )
3023, 29mtbid 291 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  /\  w  e.  ( X  \  U
) )  ->  -.  ( F `  w )  e.  ( F " U ) )
3130ralrimiva 2639 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  A. w  e.  ( X  \  U
)  -.  ( F `
 w )  e.  ( F " U
) )
32 difss 3316 . . . . 5  |-  ( X 
\  U )  C_  X
33 eleq1 2356 . . . . . . 7  |-  ( z  =  ( F `  w )  ->  (
z  e.  ( F
" U )  <->  ( F `  w )  e.  ( F " U ) ) )
3433notbid 285 . . . . . 6  |-  ( z  =  ( F `  w )  ->  ( -.  z  e.  ( F " U )  <->  -.  ( F `  w )  e.  ( F " U
) ) )
3534ralima 5774 . . . . 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 643 . . . 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 223 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  A. z  e.  ( F " ( X  \  U ) )  -.  z  e.  ( F " U ) )
38 disjr 3509 . . 3  |-  ( ( ( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) 
<-> 
A. z  e.  ( F " ( X 
\  U ) )  -.  z  e.  ( F " U ) )
3937, 38sylibr 203 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) )
40 sseq0 3499 . 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 642 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( F " U
)  i^i  ( F " ( A  \  U
) ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468    e. cmpt 4093   dom cdm 4705    |` cres 4707   "cima 4708    Fn wfn 5266   ` cfv 5271  TopOnctopon 16648
This theorem is referenced by:  kqcldsat  17440  regr1lem  17446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-topon 16655
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