Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmseu Structured version   Unicode version

Theorem cvmseu 28910
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 1001 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  B )
2 simpr3 1002 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  -> 
( F `  A
)  e.  U )
3 cvmcn 28896 . . . . . . 7  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
43adantr 463 . . . . . 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 2382 . . . . . . 7  |-  U. J  =  U. J
75, 6cnf 19833 . . . . . 6  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
8 ffn 5639 . . . . . 6  |-  ( F : B --> U. J  ->  F  Fn  B )
9 elpreima 5909 . . . . . 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 920 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  ( `' F " U ) )
12 simpr1 1000 . . . . 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 28903 . . . . 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 2473 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  A  e.  U. T )
17 eluni2 4167 . . 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 3797 . . . 4  |-  ( ( A  e.  x  /\  A  e.  z )  ->  ( x  i^i  z
)  =/=  (/) )
2013cvmsdisj 28904 . . . . . . . 8  |-  ( ( T  e.  ( S `
 U )  /\  x  e.  T  /\  z  e.  T )  ->  ( x  =  z  \/  ( x  i^i  z )  =  (/) ) )
21203expb 1195 . . . . . . 7  |-  ( ( T  e.  ( S `
 U )  /\  ( x  e.  T  /\  z  e.  T
) )  ->  (
x  =  z  \/  ( x  i^i  z
)  =  (/) ) )
2212, 21sylan 469 . . . . . 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 375 . . . . 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 2598 . . . 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 2803 . 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 2455 . . 3  |-  ( x  =  z  ->  ( A  e.  x  <->  A  e.  z ) )
2827reu4 3218 . 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 662 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 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733   E!wreu 2734   {crab 2736    \ cdif 3386    i^i cin 3388   (/)c0 3711   ~Pcpw 3927   {csn 3944   U.cuni 4163    |-> cmpt 4425   `'ccnv 4912    |` cres 4915   "cima 4916    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196   ↾t crest 14828    Cn ccn 19811   Homeochmeo 20339   CovMap ccvm 28889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-map 7340  df-top 19484  df-topon 19487  df-cn 19814  df-cvm 28890
This theorem is referenced by:  cvmsiota  28911
  Copyright terms: Public domain W3C validator