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Theorem cvmshmeo 28913
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 28907 . . . . 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 2850 . . 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 5278 . . . 4  |-  ( u  =  A  ->  ( F  |`  u )  =  ( F  |`  A ) )
9 oveq2 6304 . . . . 5  |-  ( u  =  A  ->  ( Ct  u )  =  ( Ct  A ) )
109oveq1d 6311 . . . 4  |-  ( u  =  A  ->  (
( Ct  u ) Homeo ( Jt  U ) )  =  ( ( Ct  A ) Homeo ( Jt  U ) ) )
118, 10eleq12d 2539 . . 3  |-  ( u  =  A  ->  (
( F  |`  u
)  e.  ( ( Ct  u ) Homeo ( Jt  U ) )  <->  ( F  |`  A )  e.  ( ( Ct  A ) Homeo ( Jt  U ) ) ) )
1211rspccva 3209 . 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   U.cuni 4251    |-> cmpt 4515   `'ccnv 5007    |` cres 5010   "cima 5011   ` cfv 5594  (class class class)co 6296   ↾t crest 14838   Homeochmeo 20380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299
This theorem is referenced by:  cvmsf1o  28914  cvmsss2  28916  cvmopnlem  28920  cvmliftlem8  28934
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