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Theorem cvmsf1o 24740
Description:  F, localized to an element of an even covering of  U, is a bijection. (Contributed by Mario Carneiro, 14-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
cvmsf1o  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( F  |`  A ) : A -1-1-onto-> 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 cvmsf1o
StepHypRef Expression
1 cvmtop1 24728 . . . . 5  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
213ad2ant1 978 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  C  e.  Top )
3 eqid 2389 . . . . 5  |-  U. C  =  U. C
43toptopon 16923 . . . 4  |-  ( C  e.  Top  <->  C  e.  (TopOn `  U. C ) )
52, 4sylib 189 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  C  e.  (TopOn `  U. C ) )
6 cvmcov.1 . . . . . . 7  |-  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 ) ) ) ) } )
76cvmsss 24735 . . . . . 6  |-  ( T  e.  ( S `  U )  ->  T  C_  C )
873ad2ant2 979 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  T  C_  C )
9 simp3 959 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  A  e.  T )
108, 9sseldd 3294 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  A  e.  C )
11 elssuni 3987 . . . 4  |-  ( A  e.  C  ->  A  C_ 
U. C )
1210, 11syl 16 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  A  C_ 
U. C )
13 resttopon 17149 . . 3  |-  ( ( C  e.  (TopOn `  U. C )  /\  A  C_ 
U. C )  -> 
( Ct  A )  e.  (TopOn `  A ) )
145, 12, 13syl2anc 643 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( Ct  A )  e.  (TopOn `  A ) )
15 cvmtop2 24729 . . . . 5  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
16153ad2ant1 978 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  J  e.  Top )
17 eqid 2389 . . . . 5  |-  U. J  =  U. J
1817toptopon 16923 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
1916, 18sylib 189 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  J  e.  (TopOn `  U. J ) )
206cvmsrcl 24732 . . . . 5  |-  ( T  e.  ( S `  U )  ->  U  e.  J )
21203ad2ant2 979 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  U  e.  J )
22 elssuni 3987 . . . 4  |-  ( U  e.  J  ->  U  C_ 
U. J )
2321, 22syl 16 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  U  C_ 
U. J )
24 resttopon 17149 . . 3  |-  ( ( J  e.  (TopOn `  U. J )  /\  U  C_ 
U. J )  -> 
( Jt  U )  e.  (TopOn `  U ) )
2519, 23, 24syl2anc 643 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( Jt  U )  e.  (TopOn `  U ) )
266cvmshmeo 24739 . . 3  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A )  Homeo  ( Jt  U
) ) )
27263adant1 975 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A ) 
Homeo  ( Jt  U ) ) )
28 hmeof1o2 17718 . 2  |-  ( ( ( Ct  A )  e.  (TopOn `  A )  /\  ( Jt  U )  e.  (TopOn `  U )  /\  ( F  |`  A )  e.  ( ( Ct  A ) 
Homeo  ( Jt  U ) ) )  ->  ( F  |`  A ) : A -1-1-onto-> U
)
2914, 25, 27, 28syl3anc 1184 1  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( F  |`  A ) : A -1-1-onto-> U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655    \ cdif 3262    i^i cin 3264    C_ wss 3265   (/)c0 3573   ~Pcpw 3744   {csn 3759   U.cuni 3959    e. cmpt 4209   `'ccnv 4819    |` cres 4822   "cima 4823   -1-1-onto->wf1o 5395   ` cfv 5396  (class class class)co 6022   ↾t crest 13577   Topctop 16883  TopOnctopon 16884    Homeo chmeo 17708   CovMap ccvm 24723
This theorem is referenced by:  cvmsss2  24742  cvmfolem  24747  cvmliftmolem1  24749  cvmliftlem6  24758  cvmliftlem9  24761  cvmlift2lem9a  24771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-recs 6571  df-rdg 6606  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-fin 7051  df-fi 7353  df-rest 13579  df-topgen 13596  df-top 16888  df-bases 16890  df-topon 16891  df-cn 17215  df-hmeo 17710  df-cvm 24724
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