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Theorem cvmlift2lem9a 30098
Description: Lemma for cvmlift2 30111 and cvmlift3 30123. (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.  ( ( Ct  c )
Homeo ( Jt  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.  ( ( Kt  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.
StepHypRef Expression
1 cvmlift2lem9a.f . . . 4  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
2 cvmtop1 30055 . . . 4  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
31, 2syl 17 . . 3  |-  ( ph  ->  C  e.  Top )
4 cnrest2r 20380 . . 3  |-  ( C  e.  Top  ->  (
( Kt  M )  Cn  ( Ct  W ) )  C_  ( ( Kt  M )  Cn  C ) )
53, 4syl 17 . 2  |-  ( ph  ->  ( ( Kt  M )  Cn  ( Ct  W ) )  C_  ( ( Kt  M )  Cn  C
) )
6 cvmlift2lem9a.h . . . . . 6  |-  ( ph  ->  H : Y --> B )
7 ffn 5739 . . . . . 6  |-  ( H : Y --> B  ->  H  Fn  Y )
86, 7syl 17 . . . . 5  |-  ( ph  ->  H  Fn  Y )
9 cvmlift2lem9a.4 . . . . 5  |-  ( ph  ->  M  C_  Y )
10 fnssres 5699 . . . . 5  |-  ( ( H  Fn  Y  /\  M  C_  Y )  -> 
( H  |`  M )  Fn  M )
118, 9, 10syl2anc 673 . . . 4  |-  ( ph  ->  ( H  |`  M )  Fn  M )
12 df-ima 4852 . . . . 5  |-  ( H
" M )  =  ran  ( H  |`  M )
13 cvmlift2lem9a.6 . . . . 5  |-  ( ph  ->  ( H " M
)  C_  W )
1412, 13syl5eqssr 3463 . . . 4  |-  ( ph  ->  ran  ( H  |`  M )  C_  W
)
15 df-f 5593 . . . 4  |-  ( ( H  |`  M ) : M --> W  <->  ( ( H  |`  M )  Fn  M  /\  ran  ( H  |`  M )  C_  W ) )
1611, 14, 15sylanbrc 677 . . 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 ) )
1918simpld 466 . . . . . . . . . . 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.  ( ( Ct  c )
Homeo ( Jt  k ) ) ) ) } )
2120cvmsf1o 30067 . . . . . . . . . . 11  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  A
)  /\  W  e.  T )  ->  ( F  |`  W ) : W -1-1-onto-> A )
221, 17, 19, 21syl3anc 1292 . . . . . . . . . 10  |-  ( ph  ->  ( F  |`  W ) : W -1-1-onto-> A )
2322adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( F  |`  W ) : W -1-1-onto-> A )
24 f1of1 5827 . . . . . . . . 9  |-  ( ( F  |`  W ) : W -1-1-onto-> A  ->  ( F  |`  W ) : W -1-1-> A )
2523, 24syl 17 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( F  |`  W ) : W -1-1-> A )
26 cvmlift2lem9a.b . . . . . . . . . . . 12  |-  B  = 
U. C
2726toptopon 20025 . . . . . . . . . . 11  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
283, 27sylib 201 . . . . . . . . . 10  |-  ( ph  ->  C  e.  (TopOn `  B ) )
2920cvmsss 30062 . . . . . . . . . . . . 13  |-  ( T  e.  ( S `  A )  ->  T  C_  C )
3017, 29syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  T  C_  C )
3130, 19sseldd 3419 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  C )
32 toponss 20021 . . . . . . . . . . 11  |-  ( ( C  e.  (TopOn `  B )  /\  W  e.  C )  ->  W  C_  B )
3328, 31, 32syl2anc 673 . . . . . . . . . 10  |-  ( ph  ->  W  C_  B )
34 resttopon 20254 . . . . . . . . . 10  |-  ( ( C  e.  (TopOn `  B )  /\  W  C_  B )  ->  ( Ct  W )  e.  (TopOn `  W ) )
3528, 33, 34syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  ( Ct  W )  e.  (TopOn `  W ) )
36 toponss 20021 . . . . . . . . 9  |-  ( ( ( Ct  W )  e.  (TopOn `  W )  /\  x  e.  ( Ct  W ) )  ->  x  C_  W )
3735, 36sylan 479 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  x  C_  W )
38 f1imacnv 5844 . . . . . . . 8  |-  ( ( ( F  |`  W ) : W -1-1-> A  /\  x  C_  W )  -> 
( `' ( F  |`  W ) " (
( F  |`  W )
" x ) )  =  x )
3925, 37, 38syl2anc 673 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( F  |`  W ) " (
( F  |`  W )
" x ) )  =  x )
4039imaeq2d 5174 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M ) " ( `' ( F  |`  W ) " (
( F  |`  W )
" x ) ) )  =  ( `' ( H  |`  M )
" x ) )
41 imaco 5347 . . . . . . 7  |-  ( ( `' ( H  |`  M )  o.  `' ( F  |`  W ) ) " ( ( F  |`  W ) " x ) )  =  ( `' ( H  |`  M ) " ( `' ( F  |`  W ) " ( ( F  |`  W ) " x
) ) )
42 cnvco 5025 . . . . . . . . 9  |-  `' ( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( `' ( H  |`  M )  o.  `' ( F  |`  W ) )
43 cores 5345 . . . . . . . . . . . . 13  |-  ( ran  ( H  |`  M ) 
C_  W  ->  (
( F  |`  W )  o.  ( H  |`  M ) )  =  ( F  o.  ( H  |`  M ) ) )
4414, 43syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( F  o.  ( H  |`  M ) ) )
45 resco 5346 . . . . . . . . . . . 12  |-  ( ( F  o.  H )  |`  M )  =  ( F  o.  ( H  |`  M ) )
4644, 45syl6eqr 2523 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( ( F  o.  H )  |`  M ) )
4746adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( ( F  o.  H )  |`  M ) )
4847cnveqd 5015 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  `' ( ( F  |`  W )  o.  ( H  |`  M ) )  =  `' ( ( F  o.  H )  |`  M ) )
4942, 48syl5eqr 2519 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M )  o.  `' ( F  |`  W ) )  =  `' ( ( F  o.  H
)  |`  M ) )
5049imaeq1d 5173 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( `' ( H  |`  M )  o.  `' ( F  |`  W ) ) "
( ( F  |`  W ) " x
) )  =  ( `' ( ( F  o.  H )  |`  M ) " (
( F  |`  W )
" x ) ) )
5141, 50syl5eqr 2519 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M ) " ( `' ( F  |`  W ) " (
( F  |`  W )
" x ) ) )  =  ( `' ( ( F  o.  H )  |`  M )
" ( ( F  |`  W ) " x
) ) )
5240, 51eqtr3d 2507 . . . . 5  |-  ( (
ph  /\  x  e.  ( Ct  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
5554cnrest 20378 . . . . . . . 8  |-  ( ( ( F  o.  H
)  e.  ( K  Cn  J )  /\  M  C_  Y )  -> 
( ( F  o.  H )  |`  M )  e.  ( ( Kt  M )  Cn  J ) )
5653, 9, 55syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( ( F  o.  H )  |`  M )  e.  ( ( Kt  M )  Cn  J ) )
5756adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  o.  H )  |`  M )  e.  ( ( Kt  M )  Cn  J ) )
58 resima2 5144 . . . . . . . 8  |-  ( x 
C_  W  ->  (
( F  |`  W )
" x )  =  ( F " x
) )
5937, 58syl 17 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  |`  W ) " x
)  =  ( F
" x ) )
601adantr 472 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  F  e.  ( C CovMap  J ) )
61 restopn2 20270 . . . . . . . . . 10  |-  ( ( C  e.  Top  /\  W  e.  C )  ->  ( x  e.  ( Ct  W )  <->  ( x  e.  C  /\  x  C_  W ) ) )
623, 31, 61syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( Ct  W )  <->  ( x  e.  C  /\  x  C_  W ) ) )
6362simprbda 635 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  x  e.  C )
64 cvmopn 30075 . . . . . . . 8  |-  ( ( F  e.  ( C CovMap  J )  /\  x  e.  C )  ->  ( F " x )  e.  J )
6560, 63, 64syl2anc 673 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( F " x
)  e.  J )
6659, 65eqeltrd 2549 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  |`  W ) " x
)  e.  J )
67 cnima 20358 . . . . . 6  |-  ( ( ( ( F  o.  H )  |`  M )  e.  ( ( Kt  M )  Cn  J )  /\  ( ( F  |`  W ) " x
)  e.  J )  ->  ( `' ( ( F  o.  H
)  |`  M ) "
( ( F  |`  W ) " x
) )  e.  ( Kt  M ) )
6857, 66, 67syl2anc 673 . . . . 5  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( ( F  o.  H )  |`  M ) " (
( F  |`  W )
" x ) )  e.  ( Kt  M ) )
6952, 68eqeltrd 2549 . . . 4  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M ) " x
)  e.  ( Kt  M ) )
7069ralrimiva 2809 . . 3  |-  ( ph  ->  A. x  e.  ( Ct  W ) ( `' ( H  |`  M )
" x )  e.  ( Kt  M ) )
71 cvmlift2lem9a.k . . . . . 6  |-  ( ph  ->  K  e.  Top )
7254toptopon 20025 . . . . . 6  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
7371, 72sylib 201 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
74 resttopon 20254 . . . . 5  |-  ( ( K  e.  (TopOn `  Y )  /\  M  C_  Y )  ->  ( Kt  M )  e.  (TopOn `  M ) )
7573, 9, 74syl2anc 673 . . . 4  |-  ( ph  ->  ( Kt  M )  e.  (TopOn `  M ) )
76 iscn 20328 . . . 4  |-  ( ( ( Kt  M )  e.  (TopOn `  M )  /\  ( Ct  W )  e.  (TopOn `  W ) )  -> 
( ( H  |`  M )  e.  ( ( Kt  M )  Cn  ( Ct  W ) )  <->  ( ( H  |`  M ) : M --> W  /\  A. x  e.  ( Ct  W
) ( `' ( H  |`  M ) " x )  e.  ( Kt  M ) ) ) )
7775, 35, 76syl2anc 673 . . 3  |-  ( ph  ->  ( ( H  |`  M )  e.  ( ( Kt  M )  Cn  ( Ct  W ) )  <->  ( ( H  |`  M ) : M --> W  /\  A. x  e.  ( Ct  W
) ( `' ( H  |`  M ) " x )  e.  ( Kt  M ) ) ) )
7816, 70, 77mpbir2and 936 . 2  |-  ( ph  ->  ( H  |`  M )  e.  ( ( Kt  M )  Cn  ( Ct  W ) ) )
795, 78sseldd 3419 1  |-  ( ph  ->  ( H  |`  M )  e.  ( ( Kt  M )  Cn  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760    \ cdif 3387    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   U.cuni 4190    |-> cmpt 4454   `'ccnv 4838   ran crn 4840    |` cres 4841   "cima 4842    o. ccom 4843    Fn wfn 5584   -->wf 5585   -1-1->wf1 5586   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   ↾t crest 15397   Topctop 19994  TopOnctopon 19995    Cn ccn 20317   Homeochmeo 20845   CovMap ccvm 30050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-fin 7591  df-fi 7943  df-rest 15399  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cn 20320  df-hmeo 20847  df-cvm 30051
This theorem is referenced by:  cvmlift2lem9  30106  cvmlift3lem7  30120
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