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Theorem cvmshmeo 28541
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 28535 . . . . 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 1010 . . . 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 2860 . . 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 5274 . . . 4  |-  ( u  =  A  ->  ( F  |`  u )  =  ( F  |`  A ) )
9 oveq2 6303 . . . . 5  |-  ( u  =  A  ->  ( Ct  u )  =  ( Ct  A ) )
109oveq1d 6310 . . . 4  |-  ( u  =  A  ->  (
( Ct  u ) Homeo ( Jt  U ) )  =  ( ( Ct  A ) Homeo ( Jt  U ) ) )
118, 10eleq12d 2549 . . 3  |-  ( u  =  A  ->  (
( F  |`  u
)  e.  ( ( Ct  u ) Homeo ( Jt  U ) )  <->  ( F  |`  A )  e.  ( ( Ct  A ) Homeo ( Jt  U ) ) ) )
1211rspccva 3218 . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   {crab 2821    \ cdif 3478    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   {csn 4033   U.cuni 4251    |-> cmpt 4511   `'ccnv 5004    |` cres 5007   "cima 5008   ` cfv 5594  (class class class)co 6295   ↾t crest 14693   Homeochmeo 20122
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298
This theorem is referenced by:  cvmsf1o  28542  cvmsss2  28544  cvmopnlem  28548  cvmliftlem8  28562
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