Theorem cvmlift2lem9a 31285

Description: Lemma for cvmlift2 31298 and cvmlift3 31310. (Contributed by Mario Carneiro, 9-Jul-2015.)

Hypotheses

Ref	Expression
cvmlift2lem9a.b	|- B = U. C
cvmlift2lem9a.y	|- Y = U. K
cvmlift2lem9a.s	|- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. c e. s ( A. d e. ( s \ { c } ) ( c i^i d ) = (/) /\ ( F |` c ) e. ( ( C ↾t c ) Homeo ( J ↾t k ) ) ) ) } )
cvmlift2lem9a.f	|- ( ph -> F e. ( C CovMap J ) )
cvmlift2lem9a.h	|- ( ph -> H : Y --> B )
cvmlift2lem9a.g	|- ( ph -> ( F o. H ) e. ( K Cn J ) )
cvmlift2lem9a.k	|- ( ph -> K e. Top )
cvmlift2lem9a.1	|- ( ph -> X e. Y )
cvmlift2lem9a.2	|- ( ph -> T e. ( S ` A ) )
cvmlift2lem9a.3	|- ( ph -> ( W e. T /\ ( H ` X ) e. W ) )
cvmlift2lem9a.4	|- ( ph -> M C_ Y )
cvmlift2lem9a.6	|- ( ph -> ( H " M ) C_ W )

Assertion

Ref	Expression
cvmlift2lem9a	|- ( ph -> ( H |` M ) e. ( ( K ↾t M ) Cn C ) )

Distinct variable groups:   c, d, k, s, A   F, c, d, k, s   J, c, d, k, s   T, c, d, s   C, c, d, k, s   W, c, d

Allowed substitution hints:   ph( k, s, c, d)   B( k, s, c, d)   S( k, s, c, d)   T( k)   H( k, s, c, d)   K( k, s, c, d)   M( k, s, c, d)   W( k, s)   X( k, s, c, d)   Y( k, s, c, d)

Proof of Theorem cvmlift2lem9a

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift2lem9a.f	. . . 4 |- ( ph -> F e. ( C CovMap J ) )
2		cvmtop1 31242	. . . 4 |- ( F e. ( C CovMap J ) -> C e. Top )
3	1, 2	syl 17	. . 3 |- ( ph -> C e. Top )
4		cnrest2r 21091	. . 3 |- ( C e. Top -> ( ( K ↾t M ) Cn ( C ↾t W ) ) C_ ( ( K ↾t M ) Cn C ) )
5	3, 4	syl 17	. 2 |- ( ph -> ( ( K ↾t M ) Cn ( C ↾t W ) ) C_ ( ( K ↾t M ) Cn C ) )
6		cvmlift2lem9a.h	. . . . . 6 |- ( ph -> H : Y --> B )
7		ffn 6045	. . . . . 6 |- ( H : Y --> B -> H Fn Y )
8	6, 7	syl 17	. . . . 5 |- ( ph -> H Fn Y )
9		cvmlift2lem9a.4	. . . . 5 |- ( ph -> M C_ Y )
10		fnssres 6004	. . . . 5 |- ( ( H Fn Y /\ M C_ Y ) -> ( H |` M ) Fn M )
11	8, 9, 10	syl2anc 693	. . . 4 |- ( ph -> ( H |` M ) Fn M )
12		df-ima 5127	. . . . 5 |- ( H " M ) = ran ( H |` M )
13		cvmlift2lem9a.6	. . . . 5 |- ( ph -> ( H " M ) C_ W )
14	12, 13	syl5eqssr 3650	. . . 4 |- ( ph -> ran ( H |` M ) C_ W )
15		df-f 5892	. . . 4 |- ( ( H |` M ) : M --> W <-> ( ( H |` M ) Fn M /\ ran ( H |` M ) C_ W ) )
16	11, 14, 15	sylanbrc 698	. . 3 |- ( ph -> ( H |` M ) : M --> W )
17		cvmlift2lem9a.2	. . . . . . . . . . 11 |- ( ph -> T e. ( S ` A ) )
18		cvmlift2lem9a.3	. . . . . . . . . . . 12 |- ( ph -> ( W e. T /\ ( H ` X ) e. W ) )
19	18	simpld 475	. . . . . . . . . . 11 |- ( ph -> W e. T )
20		cvmlift2lem9a.s	. . . . . . . . . . . 12 |- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. c e. s ( A. d e. ( s \ { c } ) ( c i^i d ) = (/) /\ ( F |` c ) e. ( ( C ↾t c ) Homeo ( J ↾t k ) ) ) ) } )
21	20	cvmsf1o 31254	. . . . . . . . . . 11 |- ( ( F e. ( C CovMap J ) /\ T e. ( S ` A ) /\ W e. T ) -> ( F |` W ) : W -1-1-onto-> A )
22	1, 17, 19, 21	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> ( F |` W ) : W -1-1-onto-> A )
23	22	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( F |` W ) : W -1-1-onto-> A )
24		f1of1 6136	. . . . . . . . 9 |- ( ( F |` W ) : W -1-1-onto-> A -> ( F |` W ) : W -1-1-> A )
25	23, 24	syl 17	. . . . . . . 8 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( F |` W ) : W -1-1-> A )
26		cvmlift2lem9a.b	. . . . . . . . . . . 12 |- B = U. C
27	26	toptopon 20722	. . . . . . . . . . 11 |- ( C e. Top <-> C e. (TopOn ` B ) )
28	3, 27	sylib 208	. . . . . . . . . 10 |- ( ph -> C e. (TopOn ` B ) )
29	20	cvmsss 31249	. . . . . . . . . . . . 13 |- ( T e. ( S ` A ) -> T C_ C )
30	17, 29	syl 17	. . . . . . . . . . . 12 |- ( ph -> T C_ C )
31	30, 19	sseldd 3604	. . . . . . . . . . 11 |- ( ph -> W e. C )
32		toponss 20731	. . . . . . . . . . 11 |- ( ( C e. (TopOn ` B ) /\ W e. C ) -> W C_ B )
33	28, 31, 32	syl2anc 693	. . . . . . . . . 10 |- ( ph -> W C_ B )
34		resttopon 20965	. . . . . . . . . 10 |- ( ( C e. (TopOn ` B ) /\ W C_ B ) -> ( C ↾t W ) e. (TopOn ` W ) )
35	28, 33, 34	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( C ↾t W ) e. (TopOn ` W ) )
36		toponss 20731	. . . . . . . . 9 |- ( ( ( C ↾t W ) e. (TopOn ` W ) /\ x e. ( C ↾t W ) ) -> x C_ W )
37	35, 36	sylan 488	. . . . . . . 8 |- ( ( ph /\ x e. ( C ↾t W ) ) -> x C_ W )
38		f1imacnv 6153	. . . . . . . 8 |- ( ( ( F |` W ) : W -1-1-> A /\ x C_ W ) -> ( `' ( F |` W ) " ( ( F |` W ) " x ) ) = x )
39	25, 37, 38	syl2anc 693	. . . . . . 7 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( `' ( F |` W ) " ( ( F |` W ) " x ) ) = x )
40	39	imaeq2d 5466	. . . . . 6 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( `' ( H |` M ) " ( `' ( F |` W ) " ( ( F |` W ) " x ) ) ) = ( `' ( H |` M ) " x ) )
41		imaco 5640	. . . . . . 7 |- ( ( `' ( H |` M ) o. `' ( F |` W ) ) " ( ( F |` W ) " x ) ) = ( `' ( H |` M ) " ( `' ( F |` W ) " ( ( F |` W ) " x ) ) )
42		cnvco 5308	. . . . . . . . 9 |- `' ( ( F |` W ) o. ( H |` M ) ) = ( `' ( H |` M ) o. `' ( F |` W ) )
43		cores 5638	. . . . . . . . . . . . 13 |- ( ran ( H |` M ) C_ W -> ( ( F |` W ) o. ( H |` M ) ) = ( F o. ( H |` M ) ) )
44	14, 43	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( ( F |` W ) o. ( H |` M ) ) = ( F o. ( H |` M ) ) )
45		resco 5639	. . . . . . . . . . . 12 |- ( ( F o. H ) |` M ) = ( F o. ( H |` M ) )
46	44, 45	syl6eqr 2674	. . . . . . . . . . 11 |- ( ph -> ( ( F |` W ) o. ( H |` M ) ) = ( ( F o. H ) |` M ) )
47	46	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( ( F |` W ) o. ( H |` M ) ) = ( ( F o. H ) |` M ) )
48	47	cnveqd 5298	. . . . . . . . 9 |- ( ( ph /\ x e. ( C ↾t W ) ) -> `' ( ( F |` W ) o. ( H |` M ) ) = `' ( ( F o. H ) |` M ) )
49	42, 48	syl5eqr 2670	. . . . . . . 8 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( `' ( H |` M ) o. `' ( F |` W ) ) = `' ( ( F o. H ) |` M ) )
50	49	imaeq1d 5465	. . . . . . 7 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( ( `' ( H |` M ) o. `' ( F |` W ) ) " ( ( F |` W ) " x ) ) = ( `' ( ( F o. H ) |` M ) " ( ( F |` W ) " x ) ) )
51	41, 50	syl5eqr 2670	. . . . . 6 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( `' ( H |` M ) " ( `' ( F |` W ) " ( ( F |` W ) " x ) ) ) = ( `' ( ( F o. H ) |` M ) " ( ( F |` W ) " x ) ) )
52	40, 51	eqtr3d 2658	. . . . 5 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( `' ( H |` M ) " x ) = ( `' ( ( F o. H ) |` M ) " ( ( F |` W ) " x ) ) )
53		cvmlift2lem9a.g	. . . . . . . 8 |- ( ph -> ( F o. H ) e. ( K Cn J ) )
54		cvmlift2lem9a.y	. . . . . . . . 9 |- Y = U. K
55	54	cnrest 21089	. . . . . . . 8 |- ( ( ( F o. H ) e. ( K Cn J ) /\ M C_ Y ) -> ( ( F o. H ) |` M ) e. ( ( K ↾t M ) Cn J ) )
56	53, 9, 55	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( F o. H ) |` M ) e. ( ( K ↾t M ) Cn J ) )
57	56	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( ( F o. H ) |` M ) e. ( ( K ↾t M ) Cn J ) )
58		resima2 5432	. . . . . . . 8 |- ( x C_ W -> ( ( F |` W ) " x ) = ( F " x ) )
59	37, 58	syl 17	. . . . . . 7 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( ( F |` W ) " x ) = ( F " x ) )
60	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( C ↾t W ) ) -> F e. ( C CovMap J ) )
61		restopn2 20981	. . . . . . . . . 10 |- ( ( C e. Top /\ W e. C ) -> ( x e. ( C ↾t W ) <-> ( x e. C /\ x C_ W ) ) )
62	3, 31, 61	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( x e. ( C ↾t W ) <-> ( x e. C /\ x C_ W ) ) )
63	62	simprbda 653	. . . . . . . 8 |- ( ( ph /\ x e. ( C ↾t W ) ) -> x e. C )
64		cvmopn 31262	. . . . . . . 8 |- ( ( F e. ( C CovMap J ) /\ x e. C ) -> ( F " x ) e. J )
65	60, 63, 64	syl2anc 693	. . . . . . 7 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( F " x ) e. J )
66	59, 65	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( ( F |` W ) " x ) e. J )
67		cnima 21069	. . . . . 6 |- ( ( ( ( F o. H ) |` M ) e. ( ( K ↾t M ) Cn J ) /\ ( ( F |` W ) " x ) e. J ) -> ( `' ( ( F o. H ) |` M ) " ( ( F |` W ) " x ) ) e. ( K ↾t M ) )
68	57, 66, 67	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( `' ( ( F o. H ) |` M ) " ( ( F |` W ) " x ) ) e. ( K ↾t M ) )
69	52, 68	eqeltrd 2701	. . . 4 |- ( ( ph /\ x e. ( C ↾t W ) ) -> ( `' ( H |` M ) " x ) e. ( K ↾t M ) )
70	69	ralrimiva 2966	. . 3 |- ( ph -> A. x e. ( C ↾t W ) ( `' ( H |` M ) " x ) e. ( K ↾t M ) )
71		cvmlift2lem9a.k	. . . . . 6 |- ( ph -> K e. Top )
72	54	toptopon 20722	. . . . . 6 |- ( K e. Top <-> K e. (TopOn ` Y ) )
73	71, 72	sylib 208	. . . . 5 |- ( ph -> K e. (TopOn ` Y ) )
74		resttopon 20965	. . . . 5 |- ( ( K e. (TopOn ` Y ) /\ M C_ Y ) -> ( K ↾t M ) e. (TopOn ` M ) )
75	73, 9, 74	syl2anc 693	. . . 4 |- ( ph -> ( K ↾t M ) e. (TopOn ` M ) )
76		iscn 21039	. . . 4 |- ( ( ( K ↾t M ) e. (TopOn ` M ) /\ ( C ↾t W ) e. (TopOn ` W ) ) -> ( ( H |` M ) e. ( ( K ↾t M ) Cn ( C ↾t W ) ) <-> ( ( H |` M ) : M --> W /\ A. x e. ( C ↾t W ) ( `' ( H |` M ) " x ) e. ( K ↾t M ) ) ) )
77	75, 35, 76	syl2anc 693	. . 3 |- ( ph -> ( ( H |` M ) e. ( ( K ↾t M ) Cn ( C ↾t W ) ) <-> ( ( H |` M ) : M --> W /\ A. x e. ( C ↾t W ) ( `' ( H |` M ) " x ) e. ( K ↾t M ) ) ) )
78	16, 70, 77	mpbir2and 957	. 2 |- ( ph -> ( H |` M ) e. ( ( K ↾t M ) Cn ( C ↾t W ) ) )
79	5, 78	sseldd 3604	1 |- ( ph -> ( H |` M ) e. ( ( K ↾t M ) Cn C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   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-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-rmo 2920  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-riota 6611  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:  cvmlift2lem9  31293  cvmlift3lem7  31307