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Theorem cvmcov2 31257
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. ( ( C ↾t u ) Homeo ( J ↾t 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 1061 . . 3 |- ( ( F e. ( C CovMap J ) /\ U e. J /\ P e. U ) -> F e. ( C CovMap J ) )
2 simp3 1063 . . . 4 |- ( ( F e. ( C CovMap J ) /\ U e. J /\ P e. U ) -> P e. U )
3 simp2 1062 . . . 4 |- ( ( F e. ( C CovMap J ) /\ U e. J /\ P e. U ) -> U e. J )
4 elunii 4441 . . . 4 |- ( ( P e. U /\ U e. J ) -> P e. U. J )
52, 3, 4syl2anc 693 . . 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. ( ( C ↾t u ) Homeo ( J ↾t k ) ) ) ) } )
7 eqid 2622 . . . 4 |- U. J = U. J
86, 7cvmcov 31245 . . 3 |- ( ( F e. ( C CovMap J ) /\ P e. U. J ) -> E. y e. J ( P e. y /\ ( S ` y ) =/= (/) ) )
91, 5, 8syl2anc 693 . 2 |- ( ( F e. ( C CovMap J ) /\ U e. J /\ P e. U ) -> E. y e. J ( P e. y /\ ( S ` y ) =/= (/) ) )
10 inss2 3834 . . . . 5 |- ( y i^i U ) C_ U
11 vex 3203 . . . . . . 7 |- y e. _V
1211inex1 4799 . . . . . 6 |- ( y i^i U ) e. _V
1312elpw 4164 . . . . 5 |- ( ( y i^i U ) e. ~P U <-> ( y i^i U ) C_ U )
1410, 13mpbir 221 . . . 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 804 . . . 4 |- ( ( ( F e. ( C CovMap J ) /\ U e. J /\ P e. U ) /\ ( y e. J /\ ( P e. y /\ ( S ` y ) =/= (/) ) ) ) -> P e. y )
172adantr 481 . . . 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 3798 . . 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 805 . . . 4 |- ( ( ( F e. ( C CovMap J ) /\ U e. J /\ P e. U ) /\ ( y e. J /\ ( P e. y /\ ( S ` y ) =/= (/) ) ) ) -> ( S ` y ) =/= (/) )
201adantr 481 . . . . 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 31243 . . . . . . 7 |- ( F e. ( C CovMap J ) -> J e. Top )
2220, 21syl 17 . . . . . 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 794 . . . . . 6 |- ( ( ( F e. ( C CovMap J ) /\ U e. J /\ P e. U ) /\ ( y e. J /\ ( P e. y /\ ( S ` y ) =/= (/) ) ) ) -> y e. J )
243adantr 481 . . . . . 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 20704 . . . . . 6 |- ( ( J e. Top /\ y e. J /\ U e. J ) -> ( y i^i U ) e. J )
2622, 23, 24, 25syl3anc 1326 . . . . 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 3833 . . . . . 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 31256 . . . . 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 1326 . . . 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 2690 . . . . 5 |- ( x = ( y i^i U ) -> ( P e. x <-> P e. ( y i^i U ) ) )
33 fveq2 6191 . . . . . 6 |- ( x = ( y i^i U ) -> ( S ` x ) = ( S ` ( y i^i U ) ) )
3433neeq1d 2853 . . . . 5 |- ( x = ( y i^i U ) -> ( ( S ` x ) =/= (/) <-> ( S ` ( y i^i U ) ) =/= (/) ) )
3532, 34anbi12d 747 . . . 4 |- ( x = ( y i^i U ) -> ( ( P e. x /\ ( S ` x ) =/= (/) ) <-> ( P e. ( y i^i U ) /\ ( S ` ( y i^i U ) ) =/= (/) ) ) )
3635rspcev 3309 . . 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 1324 . 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 3035 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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   "cima 5117   ` cfv 5888  (class class class)co 6650   ↾t crest 16081   Topctop 20698   Homeochmeo 21556   CovMap ccvm 31237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-hmeo 21558  df-cvm 31238
This theorem is referenced by: (None)
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