Theorem cvmsf1o 31254

Description: F, localized to an element of an even covering of U, is a bijection. (Contributed by Mario Carneiro, 14-Feb-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
cvmsf1o	|- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> ( F |` A ) : A -1-1-onto-> U )

Distinct variable groups:   k, s, u, v, C   k, F, s, u, v   k, J, s, u, v   U, k, s, u, v   T, s, u, v   u, A, v

Allowed substitution hints:   A( k, s)   S( v, u, k, s)   T( k)

Proof of Theorem cvmsf1o

Step	Hyp	Ref	Expression
1		cvmtop1 31242	. . . . 5 |- ( F e. ( C CovMap J ) -> C e. Top )
2	1	3ad2ant1 1082	. . . 4 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> C e. Top )
3		eqid 2622	. . . . 5 |- U. C = U. C
4	3	toptopon 20722	. . . 4 |- ( C e. Top <-> C e. (TopOn ` U. C ) )
5	2, 4	sylib 208	. . 3 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> C e. (TopOn ` U. C ) )
6		cvmcov.1	. . . . . . 7 |- 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	6	cvmsss 31249	. . . . . 6 |- ( T e. ( S ` U ) -> T C_ C )
8	7	3ad2ant2 1083	. . . . 5 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> T C_ C )
9		simp3 1063	. . . . 5 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> A e. T )
10	8, 9	sseldd 3604	. . . 4 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> A e. C )
11		elssuni 4467	. . . 4 |- ( A e. C -> A C_ U. C )
12	10, 11	syl 17	. . 3 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> A C_ U. C )
13		resttopon 20965	. . 3 |- ( ( C e. (TopOn ` U. C ) /\ A C_ U. C ) -> ( C ↾t A ) e. (TopOn ` A ) )
14	5, 12, 13	syl2anc 693	. 2 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> ( C ↾t A ) e. (TopOn ` A ) )
15		cvmtop2 31243	. . . . 5 |- ( F e. ( C CovMap J ) -> J e. Top )
16	15	3ad2ant1 1082	. . . 4 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> J e. Top )
17		eqid 2622	. . . . 5 |- U. J = U. J
18	17	toptopon 20722	. . . 4 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
19	16, 18	sylib 208	. . 3 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> J e. (TopOn ` U. J ) )
20	6	cvmsrcl 31246	. . . . 5 |- ( T e. ( S ` U ) -> U e. J )
21	20	3ad2ant2 1083	. . . 4 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> U e. J )
22		elssuni 4467	. . . 4 |- ( U e. J -> U C_ U. J )
23	21, 22	syl 17	. . 3 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> U C_ U. J )
24		resttopon 20965	. . 3 |- ( ( J e. (TopOn ` U. J ) /\ U C_ U. J ) -> ( J ↾t U ) e. (TopOn ` U ) )
25	19, 23, 24	syl2anc 693	. 2 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> ( J ↾t U ) e. (TopOn ` U ) )
26	6	cvmshmeo 31253	. . 3 |- ( ( T e. ( S ` U ) /\ A e. T ) -> ( F |` A ) e. ( ( C ↾t A ) Homeo ( J ↾t U ) ) )
27	26	3adant1 1079	. 2 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> ( F |` A ) e. ( ( C ↾t A ) Homeo ( J ↾t U ) ) )
28		hmeof1o2 21566	. 2 |- ( ( ( C ↾t A ) e. (TopOn ` A ) /\ ( J ↾t U ) e. (TopOn ` U ) /\ ( F |` A ) e. ( ( C ↾t A ) Homeo ( J ↾t U ) ) ) -> ( F |` A ) : A -1-1-onto-> U )
29	14, 25, 27, 28	syl3anc 1326	1 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` U ) /\ A e. T ) -> ( F |` A ) : A -1-1-onto-> U )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {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   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   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:  cvmsss2  31256  cvmfolem  31261  cvmliftmolem1  31263  cvmliftlem6  31272  cvmliftlem9  31275  cvmlift2lem9a  31285