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Theorem cvmcov2 27186
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 988 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  F  e.  ( C CovMap  J ) )
2 simp3 990 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  P  e.  U )
3 simp2 989 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  U  e.  J )
4 elunii 4117 . . . 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 2443 . . . 4  |-  U. J  =  U. J
86, 7cvmcov 27174 . . 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 3592 . . . . 5  |-  ( y  i^i  U )  C_  U
11 vex 2996 . . . . . . 7  |-  y  e. 
_V
1211inex1 4454 . . . . . 6  |-  ( y  i^i  U )  e. 
_V
1312elpw 3887 . . . . 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 3561 . . 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 27172 . . . . . . 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 18534 . . . . . 6  |-  ( ( J  e.  Top  /\  y  e.  J  /\  U  e.  J )  ->  ( y  i^i  U
)  e.  J )
2622, 23, 24, 25syl3anc 1218 . . . . 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 3591 . . . . . 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 27185 . . . . 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 1218 . . . 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 2504 . . . . 5  |-  ( x  =  ( y  i^i 
U )  ->  ( P  e.  x  <->  P  e.  ( y  i^i  U
) ) )
33 fveq2 5712 . . . . . 6  |-  ( x  =  ( y  i^i 
U )  ->  ( S `  x )  =  ( S `  ( y  i^i  U
) ) )
3433neeq1d 2641 . . . . 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 3094 . . 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 1216 . 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 2866 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737   {crab 2740    \ cdif 3346    i^i cin 3348    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   {csn 3898   U.cuni 4112    e. cmpt 4371   `'ccnv 4860    |` cres 4863   "cima 4864   ` cfv 5439  (class class class)co 6112   ↾t crest 14380   Topctop 18520   Homeochmeo 19348   CovMap ccvm 27166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-fin 7335  df-fi 7682  df-rest 14382  df-topgen 14403  df-top 18525  df-bases 18527  df-topon 18528  df-cn 18853  df-hmeo 19350  df-cvm 27167
This theorem is referenced by: (None)
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