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Theorem cvmsiota 28473
Description: Identify the unique element of  T containing  A. (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
cvmsiota.2  |-  W  =  ( iota_ x  e.  T  A  e.  x )
Assertion
Ref Expression
cvmsiota  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  -> 
( W  e.  T  /\  A  e.  W
) )
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    v, W    u, A, v, x    v, B, x
Allowed substitution hints:    A( k, s)    B( u, k, s)    S( v, u, k, s)    T( k)    W( x, u, k, s)

Proof of Theorem cvmsiota
StepHypRef Expression
1 cvmsiota.2 . . 3  |-  W  =  ( iota_ x  e.  T  A  e.  x )
2 cvmcov.1 . . . . 5  |-  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 ) ) ) ) } )
3 cvmseu.1 . . . . 5  |-  B  = 
U. C
42, 3cvmseu 28472 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  E! x  e.  T  A  e.  x )
5 riotacl2 6260 . . . 4  |-  ( E! x  e.  T  A  e.  x  ->  ( iota_ x  e.  T  A  e.  x )  e.  {
x  e.  T  |  A  e.  x }
)
64, 5syl 16 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  -> 
( iota_ x  e.  T  A  e.  x )  e.  { x  e.  T  |  A  e.  x } )
71, 6syl5eqel 2559 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  ->  W  e.  { x  e.  T  |  A  e.  x } )
8 eleq2 2540 . . 3  |-  ( v  =  W  ->  ( A  e.  v  <->  A  e.  W ) )
9 eleq2 2540 . . . 4  |-  ( x  =  v  ->  ( A  e.  x  <->  A  e.  v ) )
109cbvrabv 3112 . . 3  |-  { x  e.  T  |  A  e.  x }  =  {
v  e.  T  |  A  e.  v }
118, 10elrab2 3263 . 2  |-  ( W  e.  { x  e.  T  |  A  e.  x }  <->  ( W  e.  T  /\  A  e.  W ) )
127, 11sylib 196 1  |-  ( ( F  e.  ( C CovMap  J )  /\  ( T  e.  ( S `  U )  /\  A  e.  B  /\  ( F `  A )  e.  U ) )  -> 
( W  e.  T  /\  A  e.  W
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E!wreu 2816   {crab 2818    \ cdif 3473    i^i cin 3475   (/)c0 3785   ~Pcpw 4010   {csn 4027   U.cuni 4245    |-> cmpt 4505   `'ccnv 4998    |` cres 5001   "cima 5002   ` cfv 5588   iota_crio 6245  (class class class)co 6285   ↾t crest 14679   Homeochmeo 20081   CovMap ccvm 28451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-map 7423  df-top 19206  df-topon 19209  df-cn 19534  df-cvm 28452
This theorem is referenced by:  cvmopnlem  28474  cvmliftmolem2  28478  cvmliftlem6  28486  cvmliftlem8  28488  cvmliftlem9  28489  cvmlift2lem9  28507  cvmlift3lem6  28520  cvmlift3lem7  28521
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