Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmcov2 Structured version   Unicode version

Theorem cvmcov2 28388
Description: The covering map property can be restricted to an open subset. (Contributed by Mario Carneiro, 7-Jul-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
cvmcov2  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  E. x  e.  ~P  U ( P  e.  x  /\  ( S `  x )  =/=  (/) ) )
Distinct variable groups:    k, s, u, v, x, C    k, F, s, u, v, x    P, k, x    k, J, s, u, v, x   
x, S    U, k,
s, u, v, x
Allowed substitution hints:    P( v, u, s)    S( v, u, k, s)

Proof of Theorem cvmcov2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  F  e.  ( C CovMap  J ) )
2 simp3 998 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  P  e.  U )
3 simp2 997 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  U  e.  J )
4 elunii 4250 . . . 4  |-  ( ( P  e.  U  /\  U  e.  J )  ->  P  e.  U. J
)
52, 3, 4syl2anc 661 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  P  e.  U. J )
6 cvmcov.1 . . . 4  |-  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 ) ) ) ) } )
7 eqid 2467 . . . 4  |-  U. J  =  U. J
86, 7cvmcov 28376 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  P  e.  U. J )  ->  E. y  e.  J  ( P  e.  y  /\  ( S `  y
)  =/=  (/) ) )
91, 5, 8syl2anc 661 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  E. y  e.  J  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) )
10 inss2 3719 . . . . 5  |-  ( y  i^i  U )  C_  U
11 vex 3116 . . . . . . 7  |-  y  e. 
_V
1211inex1 4588 . . . . . 6  |-  ( y  i^i  U )  e. 
_V
1312elpw 4016 . . . . 5  |-  ( ( y  i^i  U )  e.  ~P U  <->  ( y  i^i  U )  C_  U
)
1410, 13mpbir 209 . . . 4  |-  ( y  i^i  U )  e. 
~P U
1514a1i 11 . . 3  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  ( y  i^i  U )  e.  ~P U )
16 simprrl 763 . . . 4  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  P  e.  y )
172adantr 465 . . . 4  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  P  e.  U )
1816, 17elind 3688 . . 3  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  P  e.  ( y  i^i  U
) )
19 simprrr 764 . . . 4  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  ( S `  y )  =/=  (/) )
201adantr 465 . . . . 5  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  F  e.  ( C CovMap  J ) )
21 cvmtop2 28374 . . . . . . 7  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
2220, 21syl 16 . . . . . 6  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  J  e.  Top )
23 simprl 755 . . . . . 6  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  y  e.  J )
243adantr 465 . . . . . 6  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  U  e.  J )
25 inopn 19203 . . . . . 6  |-  ( ( J  e.  Top  /\  y  e.  J  /\  U  e.  J )  ->  ( y  i^i  U
)  e.  J )
2622, 23, 24, 25syl3anc 1228 . . . . 5  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  ( y  i^i  U )  e.  J
)
27 inss1 3718 . . . . . 6  |-  ( y  i^i  U )  C_  y
2827a1i 11 . . . . 5  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  ( y  i^i  U )  C_  y
)
296cvmsss2 28387 . . . . 5  |-  ( ( F  e.  ( C CovMap  J )  /\  (
y  i^i  U )  e.  J  /\  (
y  i^i  U )  C_  y )  ->  (
( S `  y
)  =/=  (/)  ->  ( S `  ( y  i^i  U ) )  =/=  (/) ) )
3020, 26, 28, 29syl3anc 1228 . . . 4  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  ( ( S `  y )  =/=  (/)  ->  ( S `  ( y  i^i  U
) )  =/=  (/) ) )
3119, 30mpd 15 . . 3  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  ( S `  ( y  i^i  U
) )  =/=  (/) )
32 eleq2 2540 . . . . 5  |-  ( x  =  ( y  i^i 
U )  ->  ( P  e.  x  <->  P  e.  ( y  i^i  U
) ) )
33 fveq2 5866 . . . . . 6  |-  ( x  =  ( y  i^i 
U )  ->  ( S `  x )  =  ( S `  ( y  i^i  U
) ) )
3433neeq1d 2744 . . . . 5  |-  ( x  =  ( y  i^i 
U )  ->  (
( S `  x
)  =/=  (/)  <->  ( S `  ( y  i^i  U
) )  =/=  (/) ) )
3532, 34anbi12d 710 . . . 4  |-  ( x  =  ( y  i^i 
U )  ->  (
( P  e.  x  /\  ( S `  x
)  =/=  (/) )  <->  ( P  e.  ( y  i^i  U
)  /\  ( S `  ( y  i^i  U
) )  =/=  (/) ) ) )
3635rspcev 3214 . . 3  |-  ( ( ( y  i^i  U
)  e.  ~P U  /\  ( P  e.  ( y  i^i  U )  /\  ( S `  ( y  i^i  U
) )  =/=  (/) ) )  ->  E. x  e.  ~P  U ( P  e.  x  /\  ( S `
 x )  =/=  (/) ) )
3715, 18, 31, 36syl12anc 1226 . 2  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  E. x  e.  ~P  U ( P  e.  x  /\  ( S `  x )  =/=  (/) ) )
389, 37rexlimddv 2959 1  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  E. x  e.  ~P  U ( P  e.  x  /\  ( S `  x )  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {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   ` cfv 5588  (class class class)co 6284   ↾t crest 14676   Topctop 19189   Homeochmeo 20017   CovMap ccvm 28368
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 6576
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-fin 7520  df-fi 7871  df-rest 14678  df-topgen 14699  df-top 19194  df-bases 19196  df-topon 19197  df-cn 19522  df-hmeo 20019  df-cvm 28369
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator