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Theorem cvmsiota 29785
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 29784 . . . 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 6271 . . . 4  |-  ( E! x  e.  T  A  e.  x  ->  ( iota_ x  e.  T  A  e.  x )  e.  {
x  e.  T  |  A  e.  x }
)
64, 5syl 17 . . 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 2512 . 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 2493 . . 3  |-  ( v  =  W  ->  ( A  e.  v  <->  A  e.  W ) )
9 eleq2 2493 . . . 4  |-  ( x  =  v  ->  ( A  e.  x  <->  A  e.  v ) )
109cbvrabv 3077 . . 3  |-  { x  e.  T  |  A  e.  x }  =  {
v  e.  T  |  A  e.  v }
118, 10elrab2 3228 . 2  |-  ( W  e.  { x  e.  T  |  A  e.  x }  <->  ( W  e.  T  /\  A  e.  W ) )
127, 11sylib 199 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   E!wreu 2775   {crab 2777    \ cdif 3430    i^i cin 3432   (/)c0 3758   ~Pcpw 3976   {csn 3993   U.cuni 4213    |-> cmpt 4475   `'ccnv 4844    |` cres 4847   "cima 4848   ` cfv 5592   iota_crio 6257  (class class class)co 6296   ↾t crest 15271   Homeochmeo 20692   CovMap ccvm 29763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7473  df-top 19845  df-topon 19847  df-cn 20167  df-cvm 29764
This theorem is referenced by:  cvmopnlem  29786  cvmliftmolem2  29790  cvmliftlem6  29798  cvmliftlem8  29800  cvmliftlem9  29801  cvmlift2lem9  29819  cvmlift3lem6  29832  cvmlift3lem7  29833
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