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Theorem cvmsf1o 23818
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 23806 . . . . 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 2296 . . . . 5  |-  U. C  =  U. C
43toptopon 16687 . . . 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 23813 . . . . . 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 3194 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  A  e.  C )
11 elssuni 3871 . . . 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 16908 . . 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 23807 . . . . 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 2296 . . . . 5  |-  U. J  =  U. J
1817toptopon 16687 . . . 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 23810 . . . . 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 3871 . . . 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 16908 . . 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 23817 . . 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 17470 . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   U.cuni 3843    e. cmpt 4093   `'ccnv 4704    |` cres 4707   "cima 4708   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648    Homeo chmeo 17460   CovMap ccvm 23801
This theorem is referenced by:  cvmsss2  23820  cvmfolem  23825  cvmliftmolem1  23827  cvmliftlem6  23836  cvmliftlem9  23839  cvmlift2lem9a  23849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cn 16973  df-hmeo 17462  df-cvm 23802
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