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Theorem cvmsf1o 24951
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 24939 . . . . 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 2435 . . . . 5  |-  U. C  =  U. C
43toptopon 16990 . . . 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 24946 . . . . . 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 3341 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  U
)  /\  A  e.  T )  ->  A  e.  C )
11 elssuni 4035 . . . 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 17217 . . 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 24940 . . . . 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 2435 . . . . 5  |-  U. J  =  U. J
1817toptopon 16990 . . . 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 24943 . . . . 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 4035 . . . 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 17217 . . 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 24950 . . 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 17787 . 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 1652    e. wcel 1725   A.wral 2697   {crab 2701    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806   U.cuni 4007    e. cmpt 4258   `'ccnv 4869    |` cres 4872   "cima 4873   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   ↾t crest 13640   Topctop 16950  TopOnctopon 16951    Homeo chmeo 17777   CovMap ccvm 24934
This theorem is referenced by:  cvmsss2  24953  cvmfolem  24958  cvmliftmolem1  24960  cvmliftlem6  24969  cvmliftlem9  24972  cvmlift2lem9a  24982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-fin 7105  df-fi 7408  df-rest 13642  df-topgen 13659  df-top 16955  df-bases 16957  df-topon 16958  df-cn 17283  df-hmeo 17779  df-cvm 24935
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