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Theorem cvmsf1o 28357
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 28345 . . . . 5  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
213ad2ant1 1017 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  C  e.  Top )
3 eqid 2467 . . . . 5  |-  U. C  =  U. C
43toptopon 19201 . . . 4  |-  ( C  e.  Top  <->  C  e.  (TopOn `  U. C ) )
52, 4sylib 196 . . 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 28352 . . . . . 6  |-  ( T  e.  ( S `  U )  ->  T  C_  C )
873ad2ant2 1018 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  T  C_  C )
9 simp3 998 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  A  e.  T )
108, 9sseldd 3505 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  A  e.  C )
11 elssuni 4275 . . . 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 19428 . . 3  |-  ( ( C  e.  (TopOn `  U. C )  /\  A  C_ 
U. C )  -> 
( Ct  A )  e.  (TopOn `  A ) )
145, 12, 13syl2anc 661 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( Ct  A )  e.  (TopOn `  A ) )
15 cvmtop2 28346 . . . . 5  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
16153ad2ant1 1017 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  J  e.  Top )
17 eqid 2467 . . . . 5  |-  U. J  =  U. J
1817toptopon 19201 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
1916, 18sylib 196 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  J  e.  (TopOn `  U. J ) )
206cvmsrcl 28349 . . . . 5  |-  ( T  e.  ( S `  U )  ->  U  e.  J )
21203ad2ant2 1018 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  U  e.  J )
22 elssuni 4275 . . . 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 19428 . . 3  |-  ( ( J  e.  (TopOn `  U. J )  /\  U  C_ 
U. J )  -> 
( Jt  U )  e.  (TopOn `  U ) )
2519, 23, 24syl2anc 661 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( Jt  U )  e.  (TopOn `  U ) )
266cvmshmeo 28356 . . 3  |-  ( ( T  e.  ( S `
 U )  /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A ) Homeo ( Jt  U ) ) )
27263adant1 1014 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( F  |`  A )  e.  ( ( Ct  A )
Homeo ( Jt  U ) ) )
28 hmeof1o2 19999 . 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 1228 1  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  ( F  |`  A ) : A -1-1-onto-> U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   U.cuni 4245    |-> cmpt 4505   `'ccnv 4998    |` cres 5001   "cima 5002   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   ↾t crest 14672   Topctop 19161  TopOnctopon 19162   Homeochmeo 19989   CovMap ccvm 28340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-fin 7517  df-fi 7867  df-rest 14674  df-topgen 14695  df-top 19166  df-bases 19168  df-topon 19169  df-cn 19494  df-hmeo 19991  df-cvm 28341
This theorem is referenced by:  cvmsss2  28359  cvmfolem  28364  cvmliftmolem1  28366  cvmliftlem6  28375  cvmliftlem9  28378  cvmlift2lem9a  28388
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