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Theorem cvmsf1o 23803
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 23791 . . . . 5  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
213ad2ant1 976 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  C  e.  Top )
3 eqid 2283 . . . . 5  |-  U. C  =  U. C
43toptopon 16671 . . . 4  |-  ( C  e.  Top  <->  C  e.  (TopOn `  U. C ) )
52, 4sylib 188 . . 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 23798 . . . . . 6  |-  ( T  e.  ( S `  U )  ->  T  C_  C )
873ad2ant2 977 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  T  C_  C )
9 simp3 957 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  A  e.  T )
108, 9sseldd 3181 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  A  e.  C )
11 elssuni 3855 . . . 4  |-  ( A  e.  C  ->  A  C_ 
U. C )
1210, 11syl 15 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  A  C_ 
U. C )
13 resttopon 16892 . . 3  |-  ( ( C  e.  (TopOn `  U. C )  /\  A  C_ 
U. C )  -> 
( Ct  A )  e.  (TopOn `  A ) )
145, 12, 13syl2anc 642 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( Ct  A )  e.  (TopOn `  A ) )
15 cvmtop2 23792 . . . . 5  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
16153ad2ant1 976 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  J  e.  Top )
17 eqid 2283 . . . . 5  |-  U. J  =  U. J
1817toptopon 16671 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
1916, 18sylib 188 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  J  e.  (TopOn `  U. J ) )
206cvmsrcl 23795 . . . . 5  |-  ( T  e.  ( S `  U )  ->  U  e.  J )
21203ad2ant2 977 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  U  e.  J )
22 elssuni 3855 . . . 4  |-  ( U  e.  J  ->  U  C_ 
U. J )
2321, 22syl 15 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  U  C_ 
U. J )
24 resttopon 16892 . . 3  |-  ( ( J  e.  (TopOn `  U. J )  /\  U  C_ 
U. J )  -> 
( Jt  U )  e.  (TopOn `  U ) )
2519, 23, 24syl2anc 642 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( Jt  U )  e.  (TopOn `  U ) )
266cvmshmeo 23802 . . 3  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A )  Homeo  ( Jt  U
) ) )
27263adant1 973 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A ) 
Homeo  ( Jt  U ) ) )
28 hmeof1o2 17454 . 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 1182 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   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   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631  TopOnctopon 16632    Homeo chmeo 17444   CovMap ccvm 23786
This theorem is referenced by:  cvmsss2  23805  cvmfolem  23810  cvmliftmolem1  23812  cvmliftlem6  23821  cvmliftlem9  23824  cvmlift2lem9a  23834
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-hmeo 17446  df-cvm 23787
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