Theorem cvmcov 31245

Description: Property of a covering map. In order to make the covering property more manageable, we define here the set S ( k ) of all even coverings of an open set k in the range. Then the covering property states that every point has a neighborhood which has an even covering. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypotheses

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 ) ) ) ) } )
cvmcov.2	|- X = U. J

Assertion

Ref	Expression
cvmcov	|- ( ( F e. ( C CovMap J ) /\ P e. X ) -> E. x e. J ( 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   x, X

Allowed substitution hints:   P( v, u, s)   S( v, u, k, s)   X( v, u, k, s)

Proof of Theorem cvmcov

Step	Hyp	Ref	Expression
1		cvmcov.1	. . . . 5 |- 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 ) ) ) ) } )
2		cvmcov.2	. . . . 5 |- X = U. J
3	1, 2	iscvm 31241	. . . 4 |- ( F e. ( C CovMap J ) <-> ( ( C e. Top /\ J e. Top /\ F e. ( C Cn J ) ) /\ A. x e. X E. k e. J ( x e. k /\ ( S ` k ) =/= (/) ) ) )
4	3	simprbi 480	. . 3 |- ( F e. ( C CovMap J ) -> A. x e. X E. k e. J ( x e. k /\ ( S ` k ) =/= (/) ) )
5		eleq1 2689	. . . . . 6 |- ( x = P -> ( x e. k <-> P e. k ) )
6	5	anbi1d 741	. . . . 5 |- ( x = P -> ( ( x e. k /\ ( S ` k ) =/= (/) ) <-> ( P e. k /\ ( S ` k ) =/= (/) ) ) )
7	6	rexbidv 3052	. . . 4 |- ( x = P -> ( E. k e. J ( x e. k /\ ( S ` k ) =/= (/) ) <-> E. k e. J ( P e. k /\ ( S ` k ) =/= (/) ) ) )
8	7	rspcv 3305	. . 3 |- ( P e. X -> ( A. x e. X E. k e. J ( x e. k /\ ( S ` k ) =/= (/) ) -> E. k e. J ( P e. k /\ ( S ` k ) =/= (/) ) ) )
9	4, 8	mpan9 486	. 2 |- ( ( F e. ( C CovMap J ) /\ P e. X ) -> E. k e. J ( P e. k /\ ( S ` k ) =/= (/) ) )
10		nfv 1843	. . . 4 |- F/ k P e. x
11		nfmpt1 4747	. . . . . . 7 |- F/_ k ( 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 ) ) ) ) } )
12	1, 11	nfcxfr 2762	. . . . . 6 |- F/_ k S
13		nfcv 2764	. . . . . 6 |- F/_ k x
14	12, 13	nffv 6198	. . . . 5 |- F/_ k ( S ` x )
15		nfcv 2764	. . . . 5 |- F/_ k (/)
16	14, 15	nfne 2894	. . . 4 |- F/ k ( S ` x ) =/= (/)
17	10, 16	nfan 1828	. . 3 |- F/ k ( P e. x /\ ( S ` x ) =/= (/) )
18		nfv 1843	. . 3 |- F/ x ( P e. k /\ ( S ` k ) =/= (/) )
19		eleq2 2690	. . . 4 |- ( x = k -> ( P e. x <-> P e. k ) )
20		fveq2 6191	. . . . 5 |- ( x = k -> ( S ` x ) = ( S ` k ) )
21	20	neeq1d 2853	. . . 4 |- ( x = k -> ( ( S ` x ) =/= (/) <-> ( S ` k ) =/= (/) ) )
22	19, 21	anbi12d 747	. . 3 |- ( x = k -> ( ( P e. x /\ ( S ` x ) =/= (/) ) <-> ( P e. k /\ ( S ` k ) =/= (/) ) ) )
23	17, 18, 22	cbvrex 3168	. 2 |- ( E. x e. J ( P e. x /\ ( S ` x ) =/= (/) ) <-> E. k e. J ( P e. k /\ ( S ` k ) =/= (/) ) )
24	9, 23	sylibr 224	1 |- ( ( F e. ( C CovMap J ) /\ P e. X ) -> E. x e. J ( P e. x /\ ( S ` x ) =/= (/) ) )

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   (/)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   Cn ccn 21028   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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-cvm 31238

This theorem is referenced by:  cvmcov2  31257  cvmopnlem  31260  cvmfolem  31261  cvmliftmolem2  31264  cvmliftlem15  31280  cvmlift2lem10  31294  cvmlift3lem8  31308