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Theorem cvmlift2lem9a 23849
Description: Lemma for cvmlift2 23862 and cvmlift3 23874. (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 23806 . . . 4  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
31, 2syl 15 . . 3  |-  ( ph  ->  C  e.  Top )
4 cnrest2r 17031 . . 3  |-  ( C  e.  Top  ->  (
( Kt  M )  Cn  ( Ct  W ) )  C_  ( ( Kt  M )  Cn  C ) )
53, 4syl 15 . 2  |-  ( ph  ->  ( ( Kt  M )  Cn  ( Ct  W ) )  C_  ( ( Kt  M )  Cn  C
) )
6 cvmlift2lem9a.h . . . . . 6  |-  ( ph  ->  H : Y --> B )
7 ffn 5405 . . . . . 6  |-  ( H : Y --> B  ->  H  Fn  Y )
86, 7syl 15 . . . . 5  |-  ( ph  ->  H  Fn  Y )
9 cvmlift2lem9a.4 . . . . 5  |-  ( ph  ->  M  C_  Y )
10 fnssres 5373 . . . . 5  |-  ( ( H  Fn  Y  /\  M  C_  Y )  -> 
( H  |`  M )  Fn  M )
118, 9, 10syl2anc 642 . . . 4  |-  ( ph  ->  ( H  |`  M )  Fn  M )
12 df-ima 4718 . . . . 5  |-  ( H
" M )  =  ran  ( H  |`  M )
13 cvmlift2lem9a.6 . . . . 5  |-  ( ph  ->  ( H " M
)  C_  W )
1412, 13syl5eqssr 3236 . . . 4  |-  ( ph  ->  ran  ( H  |`  M )  C_  W
)
15 df-f 5275 . . . 4  |-  ( ( H  |`  M ) : M --> W  <->  ( ( H  |`  M )  Fn  M  /\  ran  ( H  |`  M )  C_  W ) )
1611, 14, 15sylanbrc 645 . . 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 445 . . . . . . . . . . 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 23818 . . . . . . . . . . 11  |-  ( ( F  e.  ( C CovMap  J )  /\  T  e.  ( S `  A
)  /\  W  e.  T )  ->  ( F  |`  W ) : W -1-1-onto-> A )
221, 17, 19, 21syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( F  |`  W ) : W -1-1-onto-> A )
2322adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( F  |`  W ) : W -1-1-onto-> A )
24 f1of1 5487 . . . . . . . . 9  |-  ( ( F  |`  W ) : W -1-1-onto-> A  ->  ( F  |`  W ) : W -1-1-> A )
2523, 24syl 15 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( F  |`  W ) : W -1-1-> A )
26 cvmlift2lem9a.b . . . . . . . . . . . 12  |-  B  = 
U. C
2726toptopon 16687 . . . . . . . . . . 11  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
283, 27sylib 188 . . . . . . . . . 10  |-  ( ph  ->  C  e.  (TopOn `  B ) )
2920cvmsss 23813 . . . . . . . . . . . . 13  |-  ( T  e.  ( S `  A )  ->  T  C_  C )
3017, 29syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  T  C_  C )
3130, 19sseldd 3194 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  C )
32 toponss 16683 . . . . . . . . . . 11  |-  ( ( C  e.  (TopOn `  B )  /\  W  e.  C )  ->  W  C_  B )
3328, 31, 32syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  W  C_  B )
34 resttopon 16908 . . . . . . . . . 10  |-  ( ( C  e.  (TopOn `  B )  /\  W  C_  B )  ->  ( Ct  W )  e.  (TopOn `  W ) )
3528, 33, 34syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( Ct  W )  e.  (TopOn `  W ) )
36 toponss 16683 . . . . . . . . 9  |-  ( ( ( Ct  W )  e.  (TopOn `  W )  /\  x  e.  ( Ct  W ) )  ->  x  C_  W )
3735, 36sylan 457 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  x  C_  W )
38 f1imacnv 5505 . . . . . . . 8  |-  ( ( ( F  |`  W ) : W -1-1-> A  /\  x  C_  W )  -> 
( `' ( F  |`  W ) " (
( F  |`  W )
" x ) )  =  x )
3925, 37, 38syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( F  |`  W ) " (
( F  |`  W )
" x ) )  =  x )
4039imaeq2d 5028 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M ) " ( `' ( F  |`  W ) " (
( F  |`  W )
" x ) ) )  =  ( `' ( H  |`  M )
" x ) )
41 imaco 5194 . . . . . . 7  |-  ( ( `' ( H  |`  M )  o.  `' ( F  |`  W ) ) " ( ( F  |`  W ) " x ) )  =  ( `' ( H  |`  M ) " ( `' ( F  |`  W ) " ( ( F  |`  W ) " x
) ) )
42 cnvco 4881 . . . . . . . . 9  |-  `' ( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( `' ( H  |`  M )  o.  `' ( F  |`  W ) )
43 cores 5192 . . . . . . . . . . . . 13  |-  ( ran  ( H  |`  M ) 
C_  W  ->  (
( F  |`  W )  o.  ( H  |`  M ) )  =  ( F  o.  ( H  |`  M ) ) )
4414, 43syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( F  o.  ( H  |`  M ) ) )
45 resco 5193 . . . . . . . . . . . 12  |-  ( ( F  o.  H )  |`  M )  =  ( F  o.  ( H  |`  M ) )
4644, 45syl6eqr 2346 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( ( F  o.  H )  |`  M ) )
4746adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  |`  W )  o.  ( H  |`  M ) )  =  ( ( F  o.  H )  |`  M ) )
4847cnveqd 4873 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  `' ( ( F  |`  W )  o.  ( H  |`  M ) )  =  `' ( ( F  o.  H )  |`  M ) )
4942, 48syl5eqr 2342 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M )  o.  `' ( F  |`  W ) )  =  `' ( ( F  o.  H
)  |`  M ) )
5049imaeq1d 5027 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( `' ( H  |`  M )  o.  `' ( F  |`  W ) ) "
( ( F  |`  W ) " x
) )  =  ( `' ( ( F  o.  H )  |`  M ) " (
( F  |`  W )
" x ) ) )
5141, 50syl5eqr 2342 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M ) " ( `' ( F  |`  W ) " (
( F  |`  W )
" x ) ) )  =  ( `' ( ( F  o.  H )  |`  M )
" ( ( F  |`  W ) " x
) ) )
5240, 51eqtr3d 2330 . . . . 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 17029 . . . . . . . 8  |-  ( ( ( F  o.  H
)  e.  ( K  Cn  J )  /\  M  C_  Y )  -> 
( ( F  o.  H )  |`  M )  e.  ( ( Kt  M )  Cn  J ) )
5653, 9, 55syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( F  o.  H )  |`  M )  e.  ( ( Kt  M )  Cn  J ) )
5756adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  o.  H )  |`  M )  e.  ( ( Kt  M )  Cn  J ) )
58 resima2 5004 . . . . . . . 8  |-  ( x 
C_  W  ->  (
( F  |`  W )
" x )  =  ( F " x
) )
5937, 58syl 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  |`  W ) " x
)  =  ( F
" x ) )
601adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  F  e.  ( C CovMap  J ) )
61 restopn2 16924 . . . . . . . . . 10  |-  ( ( C  e.  Top  /\  W  e.  C )  ->  ( x  e.  ( Ct  W )  <->  ( x  e.  C  /\  x  C_  W ) ) )
623, 31, 61syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( Ct  W )  <->  ( x  e.  C  /\  x  C_  W ) ) )
6362simprbda 606 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  ->  x  e.  C )
64 cvmopn 23826 . . . . . . . 8  |-  ( ( F  e.  ( C CovMap  J )  /\  x  e.  C )  ->  ( F " x )  e.  J )
6560, 63, 64syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( F " x
)  e.  J )
6659, 65eqeltrd 2370 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( ( F  |`  W ) " x
)  e.  J )
67 cnima 17010 . . . . . 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 642 . . . . 5  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( ( F  o.  H )  |`  M ) " (
( F  |`  W )
" x ) )  e.  ( Kt  M ) )
6952, 68eqeltrd 2370 . . . 4  |-  ( (
ph  /\  x  e.  ( Ct  W ) )  -> 
( `' ( H  |`  M ) " x
)  e.  ( Kt  M ) )
7069ralrimiva 2639 . . 3  |-  ( ph  ->  A. x  e.  ( Ct  W ) ( `' ( H  |`  M )
" x )  e.  ( Kt  M ) )
71 cvmlift2lem9a.k . . . . . 6  |-  ( ph  ->  K  e.  Top )
7254toptopon 16687 . . . . . 6  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
7371, 72sylib 188 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
74 resttopon 16908 . . . . 5  |-  ( ( K  e.  (TopOn `  Y )  /\  M  C_  Y )  ->  ( Kt  M )  e.  (TopOn `  M ) )
7573, 9, 74syl2anc 642 . . . 4  |-  ( ph  ->  ( Kt  M )  e.  (TopOn `  M ) )
76 iscn 16981 . . . 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 642 . . 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 888 . 2  |-  ( ph  ->  ( H  |`  M )  e.  ( ( Kt  M )  Cn  ( Ct  W ) ) )
795, 78sseldd 3194 1  |-  ( ph  ->  ( H  |`  M )  e.  ( ( Kt  M )  Cn  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   U.cuni 3843    e. cmpt 4093   `'ccnv 4704   ran crn 4706    |` cres 4707   "cima 4708    o. ccom 4709    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647  TopOnctopon 16648    Cn ccn 16970    Homeo chmeo 17460   CovMap ccvm 23801
This theorem is referenced by:  cvmlift2lem9  23857  cvmlift3lem7  23871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cn 16973  df-hmeo 17462  df-cvm 23802
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