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Theorem cvmshmeo 23802
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 23796 . . . . 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 969 . . . 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 449 . . 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 447 . . . 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 2618 . . 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 15 . 2  |-  ( T  e.  ( S `  U )  ->  A. u  e.  T  ( F  |`  u )  e.  ( ( Ct  u )  Homeo  ( Jt  U ) ) )
8 reseq2 4950 . . . 4  |-  ( u  =  A  ->  ( F  |`  u )  =  ( F  |`  A ) )
9 oveq2 5866 . . . . 5  |-  ( u  =  A  ->  ( Ct  u )  =  ( Ct  A ) )
109oveq1d 5873 . . . 4  |-  ( u  =  A  ->  (
( Ct  u )  Homeo  ( Jt  U ) )  =  ( ( Ct  A )  Homeo  ( Jt  U ) ) )
118, 10eleq12d 2351 . . 3  |-  ( u  =  A  ->  (
( F  |`  u
)  e.  ( ( Ct  u )  Homeo  ( Jt  U ) )  <->  ( F  |`  A )  e.  ( ( Ct  A )  Homeo  ( Jt  U ) ) ) )
1211rspccva 2883 . 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 457 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 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827    e. cmpt 4077   `'ccnv 4688    |` cres 4691   "cima 4692   ` cfv 5255  (class class class)co 5858   ↾t crest 13325    Homeo chmeo 17444
This theorem is referenced by:  cvmsf1o  23803  cvmsss2  23805  cvmopnlem  23809  cvmliftlem8  23823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861
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