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Theorem cvmseu 23807
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 962 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  B )
2 simpr3 963 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  -> 
( F `  A
)  e.  U )
3 cvmcn 23793 . . . . . . . 8  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
43adantr 451 . . . . . . 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 2283 . . . . . . . 8  |-  U. J  =  U. J
75, 6cnf 16976 . . . . . . 7  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
84, 7syl 15 . . . . . 6  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  F : B --> U. J
)
9 ffn 5389 . . . . . 6  |-  ( F : B --> U. J  ->  F  Fn  B )
10 elpreima 5645 . . . . . 6  |-  ( F  Fn  B  ->  ( A  e.  ( `' F " U )  <->  ( A  e.  B  /\  ( F `  A )  e.  U ) ) )
118, 9, 103syl 18 . . . . 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 888 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  ( `' F " U ) )
13 simpr1 961 . . . . 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 23800 . . . . 5  |-  ( T  e.  ( S `  U )  ->  U. T  =  ( `' F " U ) )
1613, 15syl 15 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  U. T  =  ( `' F " U ) )
1712, 16eleqtrrd 2360 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  U. T )
18 eluni2 3831 . . 3  |-  ( A  e.  U. T  <->  E. x  e.  T  A  e.  x )
1917, 18sylib 188 . 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 3509 . . . 4  |-  ( ( A  e.  x  /\  A  e.  z )  ->  ( x  i^i  z
)  =/=  (/) )
2114cvmsdisj 23801 . . . . . . . 8  |-  ( ( T  e.  ( S `
 U )  /\  x  e.  T  /\  z  e.  T )  ->  ( x  =  z  \/  ( x  i^i  z )  =  (/) ) )
22213expb 1152 . . . . . . 7  |-  ( ( T  e.  ( S `
 U )  /\  ( x  e.  T  /\  z  e.  T
) )  ->  (
x  =  z  \/  ( x  i^i  z
)  =  (/) ) )
2313, 22sylan 457 . . . . . 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 366 . . . . 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 2513 . . . 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 28 . . 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 2635 . 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 2344 . . 3  |-  ( x  =  z  ->  ( A  e.  x  <->  A  e.  z ) )
2928reu4 2959 . 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 645 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   E!wreu 2545   {crab 2547    \ cdif 3149    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827    e. cmpt 4077   `'ccnv 4688    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   ↾t crest 13325    Cn ccn 16954    Homeo chmeo 17444   CovMap ccvm 23786
This theorem is referenced by:  cvmsiota  23808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-cvm 23787
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