Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmshmeo Structured version   Unicode version

Theorem cvmshmeo 24960
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 24954 . . . . 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 972 . . . 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 451 . . 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 449 . . . 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 2783 . . 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 5143 . . . 4  |-  ( u  =  A  ->  ( F  |`  u )  =  ( F  |`  A ) )
9 oveq2 6091 . . . . 5  |-  ( u  =  A  ->  ( Ct  u )  =  ( Ct  A ) )
109oveq1d 6098 . . . 4  |-  ( u  =  A  ->  (
( Ct  u )  Homeo  ( Jt  U ) )  =  ( ( Ct  A )  Homeo  ( Jt  U ) ) )
118, 10eleq12d 2506 . . 3  |-  ( u  =  A  ->  (
( F  |`  u
)  e.  ( ( Ct  u )  Homeo  ( Jt  U ) )  <->  ( F  |`  A )  e.  ( ( Ct  A )  Homeo  ( Jt  U ) ) ) )
1211rspccva 3053 . 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 459 1  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A )  Homeo  ( Jt  U
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017    e. cmpt 4268   `'ccnv 4879    |` cres 4882   "cima 4883   ` cfv 5456  (class class class)co 6083   ↾t crest 13650    Homeo chmeo 17787
This theorem is referenced by:  cvmsf1o  24961  cvmsss2  24963  cvmopnlem  24967  cvmliftlem8  24981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086
  Copyright terms: Public domain W3C validator