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Theorem cvmsiota 27178
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 27177 . . . 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 6078 . . . 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 2527 . 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 2504 . . 3  |-  ( v  =  W  ->  ( A  e.  v  <->  A  e.  W ) )
9 eleq2 2504 . . . 4  |-  ( x  =  v  ->  ( A  e.  x  <->  A  e.  v ) )
109cbvrabv 2983 . . 3  |-  { x  e.  T  |  A  e.  x }  =  {
v  e.  T  |  A  e.  v }
118, 10elrab2 3131 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2727   E!wreu 2729   {crab 2731    \ cdif 3337    i^i cin 3339   (/)c0 3649   ~Pcpw 3872   {csn 3889   U.cuni 4103    e. cmpt 4362   `'ccnv 4851    |` cres 4854   "cima 4855   ` cfv 5430   iota_crio 6063  (class class class)co 6103   ↾t crest 14371   Homeochmeo 19338   CovMap ccvm 27156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-map 7228  df-top 18515  df-topon 18518  df-cn 18843  df-cvm 27157
This theorem is referenced by:  cvmopnlem  27179  cvmliftmolem2  27183  cvmliftlem6  27191  cvmliftlem8  27193  cvmliftlem9  27194  cvmlift2lem9  27212  cvmlift3lem6  27225  cvmlift3lem7  27226
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