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Theorem cvmcov2 24954
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 957 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  F  e.  ( C CovMap  J ) )
2 simp3 959 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  P  e.  U )
3 simp2 958 . . . 4  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  U  e.  J )
4 elunii 4012 . . . 4  |-  ( ( P  e.  U  /\  U  e.  J )  ->  P  e.  U. J
)
52, 3, 4syl2anc 643 . . 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 2435 . . . 4  |-  U. J  =  U. J
86, 7cvmcov 24942 . . 3  |-  ( ( F  e.  ( C CovMap  J )  /\  P  e.  U. J )  ->  E. y  e.  J  ( P  e.  y  /\  ( S `  y
)  =/=  (/) ) )
91, 5, 8syl2anc 643 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  ->  E. y  e.  J  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) )
10 inss2 3554 . . . . 5  |-  ( y  i^i  U )  C_  U
11 vex 2951 . . . . . . 7  |-  y  e. 
_V
1211inex1 4336 . . . . . 6  |-  ( y  i^i  U )  e. 
_V
1312elpw 3797 . . . . 5  |-  ( ( y  i^i  U )  e.  ~P U  <->  ( y  i^i  U )  C_  U
)
1410, 13mpbir 201 . . . 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 741 . . . 4  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  P  e.  y )
172adantr 452 . . . 4  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  P  e.  U )
18 elin 3522 . . . 4  |-  ( P  e.  ( y  i^i 
U )  <->  ( P  e.  y  /\  P  e.  U ) )
1916, 17, 18sylanbrc 646 . . 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
) )
20 simprrr 742 . . . 4  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  ( S `  y )  =/=  (/) )
211adantr 452 . . . . 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 ) )
22 cvmtop2 24940 . . . . . . 7  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
2321, 22syl 16 . . . . . 6  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  J  e.  Top )
24 simprl 733 . . . . . 6  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  y  e.  J )
253adantr 452 . . . . . 6  |-  ( ( ( F  e.  ( C CovMap  J )  /\  U  e.  J  /\  P  e.  U )  /\  ( y  e.  J  /\  ( P  e.  y  /\  ( S `  y )  =/=  (/) ) ) )  ->  U  e.  J )
26 inopn 16964 . . . . . 6  |-  ( ( J  e.  Top  /\  y  e.  J  /\  U  e.  J )  ->  ( y  i^i  U
)  e.  J )
2723, 24, 25, 26syl3anc 1184 . . . . 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
)
28 inss1 3553 . . . . . 6  |-  ( y  i^i  U )  C_  y
2928a1i 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
)
306cvmsss2 24953 . . . . 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 ) )  =/=  (/) ) )
3121, 27, 29, 30syl3anc 1184 . . . 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
) )  =/=  (/) ) )
3220, 31mpd 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
) )  =/=  (/) )
33 eleq2 2496 . . . . 5  |-  ( x  =  ( y  i^i 
U )  ->  ( P  e.  x  <->  P  e.  ( y  i^i  U
) ) )
34 fveq2 5720 . . . . . 6  |-  ( x  =  ( y  i^i 
U )  ->  ( S `  x )  =  ( S `  ( y  i^i  U
) ) )
3534neeq1d 2611 . . . . 5  |-  ( x  =  ( y  i^i 
U )  ->  (
( S `  x
)  =/=  (/)  <->  ( S `  ( y  i^i  U
) )  =/=  (/) ) )
3633, 35anbi12d 692 . . . 4  |-  ( x  =  ( y  i^i 
U )  ->  (
( P  e.  x  /\  ( S `  x
)  =/=  (/) )  <->  ( P  e.  ( y  i^i  U
)  /\  ( S `  ( y  i^i  U
) )  =/=  (/) ) ) )
3736rspcev 3044 . . 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 )  =/=  (/) ) )
3815, 19, 32, 37syl12anc 1182 . 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 )  =/=  (/) ) )
399, 38rexlimddv 2826 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {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   ` cfv 5446  (class class class)co 6073   ↾t crest 13640   Topctop 16950    Homeo chmeo 17777   CovMap ccvm 24934
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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