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Theorem cvmshmeo 27160
Description: Every element of an even covering of  U is homeomorphic to  U via  F. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypothesis
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 ) ) ) ) } )
Assertion
Ref Expression
cvmshmeo  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A ) Homeo ( Jt  U ) ) )
Distinct variable groups:    k, s, u, v, C    k, F, s, u, v    k, J, s, u, v    U, k, s, u, v    T, s, u, v    u, A, v
Allowed substitution hints:    A( k, s)    S( v, u, k, s)    T( k)

Proof of Theorem cvmshmeo
StepHypRef Expression
1 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 ) ) ) ) } )
21cvmsi 27154 . . . . 5  |-  ( T  e.  ( S `  U )  ->  ( U  e.  J  /\  ( T  C_  C  /\  T  =/=  (/) )  /\  ( U. T  =  ( `' F " U )  /\  A. u  e.  T  ( A. v  e.  ( T  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  U ) ) ) ) ) )
32simp3d 1002 . . . 4  |-  ( T  e.  ( S `  U )  ->  ( U. T  =  ( `' F " U )  /\  A. u  e.  T  ( A. v  e.  ( T  \  {
u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  U ) ) ) ) )
43simprd 463 . . 3  |-  ( T  e.  ( S `  U )  ->  A. u  e.  T  ( A. v  e.  ( T  \  { u } ) ( u  i^i  v
)  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u )
Homeo ( Jt  U ) ) ) )
5 simpr 461 . . . 4  |-  ( ( A. v  e.  ( T  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u ) Homeo ( Jt  U ) ) )  -> 
( F  |`  u
)  e.  ( ( Ct  u ) Homeo ( Jt  U ) ) )
65ralimi 2791 . . 3  |-  ( A. u  e.  T  ( A. v  e.  ( T  \  { u }
) ( u  i^i  v )  =  (/)  /\  ( F  |`  u
)  e.  ( ( Ct  u ) Homeo ( Jt  U ) ) )  ->  A. u  e.  T  ( F  |`  u )  e.  ( ( Ct  u ) Homeo ( Jt  U ) ) )
74, 6syl 16 . 2  |-  ( T  e.  ( S `  U )  ->  A. u  e.  T  ( F  |`  u )  e.  ( ( Ct  u ) Homeo ( Jt  U ) ) )
8 reseq2 5105 . . . 4  |-  ( u  =  A  ->  ( F  |`  u )  =  ( F  |`  A ) )
9 oveq2 6099 . . . . 5  |-  ( u  =  A  ->  ( Ct  u )  =  ( Ct  A ) )
109oveq1d 6106 . . . 4  |-  ( u  =  A  ->  (
( Ct  u ) Homeo ( Jt  U ) )  =  ( ( Ct  A ) Homeo ( Jt  U ) ) )
118, 10eleq12d 2511 . . 3  |-  ( u  =  A  ->  (
( F  |`  u
)  e.  ( ( Ct  u ) Homeo ( Jt  U ) )  <->  ( F  |`  A )  e.  ( ( Ct  A ) Homeo ( Jt  U ) ) ) )
1211rspccva 3072 . 2  |-  ( ( A. u  e.  T  ( F  |`  u )  e.  ( ( Ct  u ) Homeo ( Jt  U ) )  /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A )
Homeo ( Jt  U ) ) )
137, 12sylan 471 1  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A ) Homeo ( Jt  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   {crab 2719    \ cdif 3325    i^i cin 3327    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   {csn 3877   U.cuni 4091    e. cmpt 4350   `'ccnv 4839    |` cres 4842   "cima 4843   ` cfv 5418  (class class class)co 6091   ↾t crest 14359   Homeochmeo 19326
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 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094
This theorem is referenced by:  cvmsf1o  27161  cvmsss2  27163  cvmopnlem  27167  cvmliftlem8  27181
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