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Theorem cvmseu 24963
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 964 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  B )
2 simpr3 965 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  -> 
( F `  A
)  e.  U )
3 cvmcn 24949 . . . . . . . 8  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
43adantr 452 . . . . . . 7  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  F  e.  ( C  Cn  J ) )
5 cvmseu.1 . . . . . . . 8  |-  B  = 
U. C
6 eqid 2436 . . . . . . . 8  |-  U. J  =  U. J
75, 6cnf 17310 . . . . . . 7  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
84, 7syl 16 . . . . . 6  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  F : B --> U. J
)
9 ffn 5591 . . . . . 6  |-  ( F : B --> U. J  ->  F  Fn  B )
10 elpreima 5850 . . . . . 6  |-  ( F  Fn  B  ->  ( A  e.  ( `' F " U )  <->  ( A  e.  B  /\  ( F `  A )  e.  U ) ) )
118, 9, 103syl 19 . . . . 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 ) ) )
121, 2, 11mpbir2and 889 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  ( `' F " U ) )
13 simpr1 963 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  T  e.  ( S `  U ) )
14 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 ) ) ) ) } )
1514cvmsuni 24956 . . . . 5  |-  ( T  e.  ( S `  U )  ->  U. T  =  ( `' F " U ) )
1613, 15syl 16 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  U. T  =  ( `' F " U ) )
1712, 16eleqtrrd 2513 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  U. T )
18 eluni2 4019 . . 3  |-  ( A  e.  U. T  <->  E. x  e.  T  A  e.  x )
1917, 18sylib 189 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  E. x  e.  T  A  e.  x )
20 inelcm 3682 . . . 4  |-  ( ( A  e.  x  /\  A  e.  z )  ->  ( x  i^i  z
)  =/=  (/) )
2114cvmsdisj 24957 . . . . . . . 8  |-  ( ( T  e.  ( S `
 U )  /\  x  e.  T  /\  z  e.  T )  ->  ( x  =  z  \/  ( x  i^i  z )  =  (/) ) )
22213expb 1154 . . . . . . 7  |-  ( ( T  e.  ( S `
 U )  /\  ( x  e.  T  /\  z  e.  T
) )  ->  (
x  =  z  \/  ( x  i^i  z
)  =  (/) ) )
2313, 22sylan 458 . . . . . 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 )  =  (/) ) )
2423ord 367 . . . . 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 )  =  (/) ) )
2524necon1ad 2671 . . . 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 ) )
2620, 25syl5 30 . . 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 ) )
2726ralrimivva 2798 . 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 ) )
28 eleq2 2497 . . 3  |-  ( x  =  z  ->  ( A  e.  x  <->  A  e.  z ) )
2928reu4 3128 . 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 ) ) )
3019, 27, 29sylanbrc 646 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 set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   E!wreu 2707   {crab 2709    \ cdif 3317    i^i cin 3319   (/)c0 3628   ~Pcpw 3799   {csn 3814   U.cuni 4015    e. cmpt 4266   `'ccnv 4877    |` cres 4880   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   ↾t crest 13648    Cn ccn 17288    Homeo chmeo 17785   CovMap ccvm 24942
This theorem is referenced by:  cvmsiota  24964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-top 16963  df-topon 16966  df-cn 17291  df-cvm 24943
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