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Theorem cvmlift2lem9a 27122
Description: Lemma for cvmlift2 27135 and cvmlift3 27147. (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 27079 . . . 4  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
31, 2syl 16 . . 3  |-  ( ph  ->  C  e.  Top )
4 cnrest2r 18850 . . 3  |-  ( C  e.  Top  ->  (
( Kt  M )  Cn  ( Ct  W ) )  C_  ( ( Kt  M )  Cn  C ) )
53, 4syl 16 . 2  |-  ( ph  ->  ( ( Kt  M )  Cn  ( Ct  W ) )  C_  ( ( Kt  M )  Cn  C
) )
6 cvmlift2lem9a.h . . . . . 6  |-  ( ph  ->  H : Y --> B )
7 ffn 5556 . . . . . 6  |-  ( H : Y --> B  ->  H  Fn  Y )
86, 7syl 16 . . . . 5  |-  ( ph  ->  H  Fn  Y )
9 cvmlift2lem9a.4 . . . . 5  |-  ( ph  ->  M  C_  Y )
10 fnssres 5521 . . . . 5  |-  ( ( H  Fn  Y  /\  M  C_  Y )  -> 
( H  |`  M )  Fn  M )
118, 9, 10syl2anc 656 . . . 4  |-  ( ph  ->  ( H  |`  M )  Fn  M )
12 df-ima 4849 . . . . 5  |-  ( H
" M )  =  ran  ( H  |`  M )
13 cvmlift2lem9a.6 . . . . 5  |-  ( ph  ->  ( H " M
)  C_  W )
1412, 13syl5eqssr 3398 . . . 4  |-  ( ph  ->  ran  ( H  |`  M )  C_  W
)
15 df-f 5419 . . . 4  |-  ( ( H  |`  M ) : M --> W  <->  ( ( H  |`  M )  Fn  M  /\  ran  ( H  |`  M )  C_  W ) )
1611, 14, 15sylanbrc 659 . . 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 456 . . . . . . . . . . 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 27091 . . . . . . . . . . 11  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  A
)  /\  W  e.  T )  ->  ( F  |`  W ) : W -1-1-onto-> A )
221, 17, 19, 21syl3anc 1213 . . . . . . . . . 10  |-  ( ph  ->  ( F  |`  W ) : W -1-1-onto-> A )
2322adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( F  |`  W ) : W -1-1-onto-> A )
24 f1of1 5637 . . . . . . . . 9  |-  ( ( F  |`  W ) : W -1-1-onto-> A  ->  ( F  |`  W ) : W -1-1-> A )
2523, 24syl 16 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( F  |`  W ) : W -1-1-> A )
26 cvmlift2lem9a.b . . . . . . . . . . . 12  |-  B  = 
U. C
2726toptopon 18497 . . . . . . . . . . 11  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
283, 27sylib 196 . . . . . . . . . 10  |-  ( ph  ->  C  e.  (TopOn `  B ) )
2920cvmsss 27086 . . . . . . . . . . . . 13  |-  ( T  e.  ( S `  A )  ->  T  C_  C )
3017, 29syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  T  C_  C )
3130, 19sseldd 3354 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  C )
32 toponss 18493 . . . . . . . . . . 11  |-  ( ( C  e.  (TopOn `  B )  /\  W  e.  C )  ->  W  C_  B )
3328, 31, 32syl2anc 656 . . . . . . . . . 10  |-  ( ph  ->  W  C_  B )
34 resttopon 18724 . . . . . . . . . 10  |-  ( ( C  e.  (TopOn `  B )  /\  W  C_  B )  ->  ( Ct  W )  e.  (TopOn `  W ) )
3528, 33, 34syl2anc 656 . . . . . . . . 9  |-  ( ph  ->  ( Ct  W )  e.  (TopOn `  W ) )
36 toponss 18493 . . . . . . . . 9  |-  ( ( ( Ct  W )  e.  (TopOn `  W )  /\  x  e.  ( Ct  W ) )  ->  x  C_  W )
3735, 36sylan 468 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  x  C_  W )
38 f1imacnv 5654 . . . . . . . 8  |-  ( ( ( F  |`  W ) : W -1-1-> A  /\  x  C_  W )  -> 
( `' ( F  |`  W ) " (
( F  |`  W )
" x ) )  =  x )
3925, 37, 38syl2anc 656 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( F  |`  W ) " (
( F  |`  W )
" x ) )  =  x )
4039imaeq2d 5166 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M ) " ( `' ( F  |`  W ) " (
( F  |`  W )
" x ) ) )  =  ( `' ( H  |`  M )
" x ) )
41 imaco 5340 . . . . . . 7  |-  ( ( `' ( H  |`  M )  o.  `' ( F  |`  W ) ) " ( ( F  |`  W ) " x ) )  =  ( `' ( H  |`  M ) " ( `' ( F  |`  W ) " ( ( F  |`  W ) " x
) ) )
42 cnvco 5021 . . . . . . . . 9  |-  `' ( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( `' ( H  |`  M )  o.  `' ( F  |`  W ) )
43 cores 5338 . . . . . . . . . . . . 13  |-  ( ran  ( H  |`  M ) 
C_  W  ->  (
( F  |`  W )  o.  ( H  |`  M ) )  =  ( F  o.  ( H  |`  M ) ) )
4414, 43syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( F  o.  ( H  |`  M ) ) )
45 resco 5339 . . . . . . . . . . . 12  |-  ( ( F  o.  H )  |`  M )  =  ( F  o.  ( H  |`  M ) )
4644, 45syl6eqr 2491 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( ( F  o.  H )  |`  M ) )
4746adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( ( F  o.  H )  |`  M ) )
4847cnveqd 5011 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  `' ( ( F  |`  W )  o.  ( H  |`  M ) )  =  `' ( ( F  o.  H )  |`  M ) )
4942, 48syl5eqr 2487 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M )  o.  `' ( F  |`  W ) )  =  `' ( ( F  o.  H
)  |`  M ) )
5049imaeq1d 5165 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( `' ( H  |`  M )  o.  `' ( F  |`  W ) ) "
( ( F  |`  W ) " x
) )  =  ( `' ( ( F  o.  H )  |`  M ) " (
( F  |`  W )
" x ) ) )
5141, 50syl5eqr 2487 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M ) " ( `' ( F  |`  W ) " (
( F  |`  W )
" x ) ) )  =  ( `' ( ( F  o.  H )  |`  M )
" ( ( F  |`  W ) " x
) ) )
5240, 51eqtr3d 2475 . . . . 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 18848 . . . . . . . 8  |-  ( ( ( F  o.  H
)  e.  ( K  Cn  J )  /\  M  C_  Y )  -> 
( ( F  o.  H )  |`  M )  e.  ( ( Kt  M )  Cn  J ) )
5653, 9, 55syl2anc 656 . . . . . . 7  |-  ( ph  ->  ( ( F  o.  H )  |`  M )  e.  ( ( Kt  M )  Cn  J ) )
5756adantr 462 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  o.  H )  |`  M )  e.  ( ( Kt  M )  Cn  J ) )
58 resima2 5140 . . . . . . . 8  |-  ( x 
C_  W  ->  (
( F  |`  W )
" x )  =  ( F " x
) )
5937, 58syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  |`  W ) " x
)  =  ( F
" x ) )
601adantr 462 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  F  e.  ( C CovMap  J ) )
61 restopn2 18740 . . . . . . . . . 10  |-  ( ( C  e.  Top  /\  W  e.  C )  ->  ( x  e.  ( Ct  W )  <->  ( x  e.  C  /\  x  C_  W ) ) )
623, 31, 61syl2anc 656 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( Ct  W )  <->  ( x  e.  C  /\  x  C_  W ) ) )
6362simprbda 620 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  x  e.  C )
64 cvmopn 27099 . . . . . . . 8  |-  ( ( F  e.  ( C CovMap  J )  /\  x  e.  C )  ->  ( F " x )  e.  J )
6560, 63, 64syl2anc 656 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( F " x
)  e.  J )
6659, 65eqeltrd 2515 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  |`  W ) " x
)  e.  J )
67 cnima 18828 . . . . . 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 656 . . . . 5  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( ( F  o.  H )  |`  M ) " (
( F  |`  W )
" x ) )  e.  ( Kt  M ) )
6952, 68eqeltrd 2515 . . . 4  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M ) " x
)  e.  ( Kt  M ) )
7069ralrimiva 2797 . . 3  |-  ( ph  ->  A. x  e.  ( Ct  W ) ( `' ( H  |`  M )
" x )  e.  ( Kt  M ) )
71 cvmlift2lem9a.k . . . . . 6  |-  ( ph  ->  K  e.  Top )
7254toptopon 18497 . . . . . 6  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
7371, 72sylib 196 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
74 resttopon 18724 . . . . 5  |-  ( ( K  e.  (TopOn `  Y )  /\  M  C_  Y )  ->  ( Kt  M )  e.  (TopOn `  M ) )
7573, 9, 74syl2anc 656 . . . 4  |-  ( ph  ->  ( Kt  M )  e.  (TopOn `  M ) )
76 iscn 18798 . . . 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 656 . . 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 908 . 2  |-  ( ph  ->  ( H  |`  M )  e.  ( ( Kt  M )  Cn  ( Ct  W ) ) )
795, 78sseldd 3354 1  |-  ( ph  ->  ( H  |`  M )  e.  ( ( Kt  M )  Cn  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   {crab 2717    \ cdif 3322    i^i cin 3324    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   {csn 3874   U.cuni 4088    e. cmpt 4347   `'ccnv 4835   ran crn 4837    |` cres 4838   "cima 4839    o. ccom 4840    Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   ↾t crest 14355   Topctop 18457  TopOnctopon 18458    Cn ccn 18787   Homeochmeo 19285   CovMap ccvm 27074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-fin 7310  df-fi 7657  df-rest 14357  df-topgen 14378  df-top 18462  df-bases 18464  df-topon 18465  df-cn 18790  df-hmeo 19287  df-cvm 27075
This theorem is referenced by:  cvmlift2lem9  27130  cvmlift3lem7  27144
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