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Theorem cvmseu 27310
Description: Every element in  U. T is a member of a unique element of  T. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypotheses
Ref Expression
cvmcov.1  |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/)
} )  |  ( U. s  =  ( `' F " k )  /\  A. u  e.  s  ( A. v  e.  ( s  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  k ) ) ) ) } )
cvmseu.1  |-  B  = 
U. C
Assertion
Ref Expression
cvmseu  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  E! x  e.  T  A  e.  x )
Distinct variable groups:    k, s, u, v, x, C    k, F, s, u, v, x   
k, J, s, u, v, x    x, S    U, k, s, u, v, x    T, s, u, v, x    u, A, v, x    v, B, x
Allowed substitution hints:    A( k, s)    B( u, k, s)    S( v, u, k, s)    T( k)

Proof of Theorem cvmseu
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpr2 995 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  B )
2 simpr3 996 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  -> 
( F `  A
)  e.  U )
3 cvmcn 27296 . . . . . . 7  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
43adantr 465 . . . . . 6  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  F  e.  ( C  Cn  J ) )
5 cvmseu.1 . . . . . . 7  |-  B  = 
U. C
6 eqid 2454 . . . . . . 7  |-  U. J  =  U. J
75, 6cnf 18983 . . . . . 6  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
8 ffn 5668 . . . . . 6  |-  ( F : B --> U. J  ->  F  Fn  B )
9 elpreima 5933 . . . . . 6  |-  ( F  Fn  B  ->  ( A  e.  ( `' F " U )  <->  ( A  e.  B  /\  ( F `  A )  e.  U ) ) )
104, 7, 8, 94syl 21 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  -> 
( A  e.  ( `' F " U )  <-> 
( A  e.  B  /\  ( F `  A
)  e.  U ) ) )
111, 2, 10mpbir2and 913 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  ( `' F " U ) )
12 simpr1 994 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  T  e.  ( S `  U ) )
13 cvmcov.1 . . . . . 6  |-  S  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/)
} )  |  ( U. s  =  ( `' F " k )  /\  A. u  e.  s  ( A. v  e.  ( s  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  k ) ) ) ) } )
1413cvmsuni 27303 . . . . 5  |-  ( T  e.  ( S `  U )  ->  U. T  =  ( `' F " U ) )
1512, 14syl 16 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  U. T  =  ( `' F " U ) )
1611, 15eleqtrrd 2545 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  U. T )
17 eluni2 4204 . . 3  |-  ( A  e.  U. T  <->  E. x  e.  T  A  e.  x )
1816, 17sylib 196 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  E. x  e.  T  A  e.  x )
19 inelcm 3842 . . . 4  |-  ( ( A  e.  x  /\  A  e.  z )  ->  ( x  i^i  z
)  =/=  (/) )
2013cvmsdisj 27304 . . . . . . . 8  |-  ( ( T  e.  ( S `
 U )  /\  x  e.  T  /\  z  e.  T )  ->  ( x  =  z  \/  ( x  i^i  z )  =  (/) ) )
21203expb 1189 . . . . . . 7  |-  ( ( T  e.  ( S `
 U )  /\  ( x  e.  T  /\  z  e.  T
) )  ->  (
x  =  z  \/  ( x  i^i  z
)  =  (/) ) )
2212, 21sylan 471 . . . . . 6  |-  ( ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A
)  e.  U ) )  /\  ( x  e.  T  /\  z  e.  T ) )  -> 
( x  =  z  \/  ( x  i^i  z )  =  (/) ) )
2322ord 377 . . . . 5  |-  ( ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A
)  e.  U ) )  /\  ( x  e.  T  /\  z  e.  T ) )  -> 
( -.  x  =  z  ->  ( x  i^i  z )  =  (/) ) )
2423necon1ad 2668 . . . 4  |-  ( ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A
)  e.  U ) )  /\  ( x  e.  T  /\  z  e.  T ) )  -> 
( ( x  i^i  z )  =/=  (/)  ->  x  =  z ) )
2519, 24syl5 32 . . 3  |-  ( ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A
)  e.  U ) )  /\  ( x  e.  T  /\  z  e.  T ) )  -> 
( ( A  e.  x  /\  A  e.  z )  ->  x  =  z ) )
2625ralrimivva 2914 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A. x  e.  T  A. z  e.  T  ( ( A  e.  x  /\  A  e.  z )  ->  x  =  z ) )
27 eleq2 2527 . . 3  |-  ( x  =  z  ->  ( A  e.  x  <->  A  e.  z ) )
2827reu4 3260 . 2  |-  ( E! x  e.  T  A  e.  x  <->  ( E. x  e.  T  A  e.  x  /\  A. x  e.  T  A. z  e.  T  ( ( A  e.  x  /\  A  e.  z )  ->  x  =  z ) ) )
2918, 26, 28sylanbrc 664 1  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  E! x  e.  T  A  e.  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   E!wreu 2801   {crab 2803    \ cdif 3434    i^i cin 3436   (/)c0 3746   ~Pcpw 3969   {csn 3986   U.cuni 4200    |-> cmpt 4459   `'ccnv 4948    |` cres 4951   "cima 4952    Fn wfn 5522   -->wf 5523   ` cfv 5527  (class class class)co 6201   ↾t crest 14479    Cn ccn 18961   Homeochmeo 19459   CovMap ccvm 27289
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 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-map 7327  df-top 18636  df-topon 18639  df-cn 18964  df-cvm 27290
This theorem is referenced by:  cvmsiota  27311
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