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Theorem cvmlift2lem11 23859
Description: Lemma for cvmlift2 23862. (Contributed by Mario Carneiro, 1-Jun-2015.)
Hypotheses
Ref Expression
cvmlift2.b  |-  B  = 
U. C
cvmlift2.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmlift2.g  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
cvmlift2.p  |-  ( ph  ->  P  e.  B )
cvmlift2.i  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
cvmlift2.h  |-  H  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( f `
 0 )  =  P ) )
cvmlift2.k  |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
f `  0 )  =  ( H `  x ) ) ) `
 y ) )
cvmlift2.m  |-  M  =  { z  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) }
cvmlift2lem11.1  |-  ( ph  ->  U  e.  II )
cvmlift2lem11.2  |-  ( ph  ->  V  e.  II )
cvmlift2lem11.3  |-  ( ph  ->  Y  e.  V )
cvmlift2lem11.4  |-  ( ph  ->  Z  e.  V )
cvmlift2lem11.5  |-  ( ph  ->  ( E. w  e.  V  ( K  |`  ( U  X.  { w } ) )  e.  ( ( ( II 
tX  II )t  ( U  X.  { w }
) )  Cn  C
)  ->  ( K  |`  ( U  X.  V
) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C ) ) )
Assertion
Ref Expression
cvmlift2lem11  |-  ( ph  ->  ( ( U  X.  { Y } )  C_  M  ->  ( U  X.  { Z } )  C_  M ) )
Distinct variable groups:    w, f, x, y, z, F    ph, f, w, x, y, z    x, M, y, z    f, J, w, x, y, z   
w, U, z    f, G, w, x, y, z   
w, V    f, H, w, x, y, z    z, Z    C, f, w, x, y, z    P, f, x, y, z    w, B, x, y, z    f, Y, w, x, y, z   
f, K, w, x, y, z
Allowed substitution hints:    B( f)    P( w)    U( x, y, f)    M( w, f)    V( x, y, z, f)    Z( x, y, w, f)

Proof of Theorem cvmlift2lem11
StepHypRef Expression
1 cvmlift2lem11.1 . . . . . . 7  |-  ( ph  ->  U  e.  II )
21adantr 451 . . . . . 6  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  U  e.  II )
3 elssuni 3871 . . . . . . 7  |-  ( U  e.  II  ->  U  C_ 
U. II )
4 iiuni 18401 . . . . . . 7  |-  ( 0 [,] 1 )  = 
U. II
53, 4syl6sseqr 3238 . . . . . 6  |-  ( U  e.  II  ->  U  C_  ( 0 [,] 1
) )
62, 5syl 15 . . . . 5  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  U  C_  ( 0 [,] 1 ) )
7 cvmlift2lem11.4 . . . . . . . 8  |-  ( ph  ->  Z  e.  V )
8 cvmlift2lem11.2 . . . . . . . 8  |-  ( ph  ->  V  e.  II )
9 elunii 3848 . . . . . . . . 9  |-  ( ( Z  e.  V  /\  V  e.  II )  ->  Z  e.  U. II )
109, 4syl6eleqr 2387 . . . . . . . 8  |-  ( ( Z  e.  V  /\  V  e.  II )  ->  Z  e.  ( 0 [,] 1 ) )
117, 8, 10syl2anc 642 . . . . . . 7  |-  ( ph  ->  Z  e.  ( 0 [,] 1 ) )
1211adantr 451 . . . . . 6  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Z  e.  ( 0 [,] 1 ) )
1312snssd 3776 . . . . 5  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Z }  C_  (
0 [,] 1 ) )
14 xpss12 4808 . . . . 5  |-  ( ( U  C_  ( 0 [,] 1 )  /\  { Z }  C_  (
0 [,] 1 ) )  ->  ( U  X.  { Z } ) 
C_  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
156, 13, 14syl2anc 642 . . . 4  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
16 cvmlift2lem11.3 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  V )
1716adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Y  e.  V )
18 cvmlift2.b . . . . . . . . . . . . 13  |-  B  = 
U. C
19 cvmlift2.f . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
20 cvmlift2.g . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
21 cvmlift2.p . . . . . . . . . . . . 13  |-  ( ph  ->  P  e.  B )
22 cvmlift2.i . . . . . . . . . . . . 13  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
23 cvmlift2.h . . . . . . . . . . . . 13  |-  H  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( f `
 0 )  =  P ) )
24 cvmlift2.k . . . . . . . . . . . . 13  |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
f `  0 )  =  ( H `  x ) ) ) `
 y ) )
2518, 19, 20, 21, 22, 23, 24cvmlift2lem5 23853 . . . . . . . . . . . 12  |-  ( ph  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B )
2625adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
278adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  V  e.  II )
28 elssuni 3871 . . . . . . . . . . . . . . . 16  |-  ( V  e.  II  ->  V  C_ 
U. II )
2928, 4syl6sseqr 3238 . . . . . . . . . . . . . . 15  |-  ( V  e.  II  ->  V  C_  ( 0 [,] 1
) )
3027, 29syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  V  C_  ( 0 [,] 1 ) )
3130, 17sseldd 3194 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Y  e.  ( 0 [,] 1 ) )
3231snssd 3776 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Y }  C_  (
0 [,] 1 ) )
33 xpss12 4808 . . . . . . . . . . . 12  |-  ( ( U  C_  ( 0 [,] 1 )  /\  { Y }  C_  (
0 [,] 1 ) )  ->  ( U  X.  { Y } ) 
C_  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
346, 32, 33syl2anc 642 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Y } )  C_  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
35 fssres 5424 . . . . . . . . . . 11  |-  ( ( K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B  /\  ( U  X.  { Y } )  C_  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  ->  ( K  |`  ( U  X.  { Y } ) ) : ( U  X.  { Y } ) --> B )
3626, 34, 35syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  { Y }
) ) : ( U  X.  { Y } ) --> B )
3734adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  ( U  X.  { Y }
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
38 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  z  e.  ( U  X.  { Y } ) )
39 simpr 447 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Y } )  C_  M
)
40 cvmlift2.m . . . . . . . . . . . . . . 15  |-  M  =  { z  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) }
4139, 40syl6sseq 3237 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Y } )  C_  { z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) } )
42 ssrab 3264 . . . . . . . . . . . . . . 15  |-  ( ( U  X.  { Y } )  C_  { z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) }  <->  ( ( U  X.  { Y }
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) )  /\  A. z  e.  ( U  X.  { Y }
) K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) ) )
4342simprbi 450 . . . . . . . . . . . . . 14  |-  ( ( U  X.  { Y } )  C_  { z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) }  ->  A. z  e.  ( U  X.  { Y } ) K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
4441, 43syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  A. z  e.  ( U  X.  { Y }
) K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) )
4544r19.21bi 2654 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
46 iitopon 18399 . . . . . . . . . . . . . . 15  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
47 txtopon 17302 . . . . . . . . . . . . . . 15  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  II  e.  (TopOn `  ( 0 [,] 1 ) ) )  ->  ( II  tX  II )  e.  (TopOn `  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
4846, 46, 47mp2an 653 . . . . . . . . . . . . . 14  |-  ( II 
tX  II )  e.  (TopOn `  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
4948toponunii 16686 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
5049cnpresti 17032 . . . . . . . . . . . 12  |-  ( ( ( U  X.  { Y } )  C_  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) )  /\  z  e.  ( U  X.  { Y } )  /\  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )  ->  ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  CnP 
C ) `  z
) )
5137, 38, 45, 50syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  CnP 
C ) `  z
) )
5251ralrimiva 2639 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  A. z  e.  ( U  X.  { Y }
) ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  CnP  C
) `  z )
)
53 resttopon 16908 . . . . . . . . . . . 12  |-  ( ( ( II  tX  II )  e.  (TopOn `  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  /\  ( U  X.  { Y }
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  ->  ( ( II 
tX  II )t  ( U  X.  { Y }
) )  e.  (TopOn `  ( U  X.  { Y } ) ) )
5448, 34, 53sylancr 644 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( ( II  tX  II )t  ( U  X.  { Y } ) )  e.  (TopOn `  ( U  X.  { Y }
) ) )
55 cvmtop1 23806 . . . . . . . . . . . . . 14  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
5619, 55syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  Top )
5756adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  C  e.  Top )
5818toptopon 16687 . . . . . . . . . . . 12  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
5957, 58sylib 188 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  C  e.  (TopOn `  B
) )
60 cncnp 17025 . . . . . . . . . . 11  |-  ( ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  e.  (TopOn `  ( U  X.  { Y }
) )  /\  C  e.  (TopOn `  B )
)  ->  ( ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  Cn  C
)  <->  ( ( K  |`  ( U  X.  { Y } ) ) : ( U  X.  { Y } ) --> B  /\  A. z  e.  ( U  X.  { Y }
) ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  CnP  C
) `  z )
) ) )
6154, 59, 60syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( II 
tX  II )t  ( U  X.  { Y }
) )  Cn  C
)  <->  ( ( K  |`  ( U  X.  { Y } ) ) : ( U  X.  { Y } ) --> B  /\  A. z  e.  ( U  X.  { Y }
) ( K  |`  ( U  X.  { Y } ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  CnP  C
) `  z )
) ) )
6236, 52, 61mpbir2and 888 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  { Y }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  Cn  C ) )
63 sneq 3664 . . . . . . . . . . . . 13  |-  ( w  =  Y  ->  { w }  =  { Y } )
6463xpeq2d 4729 . . . . . . . . . . . 12  |-  ( w  =  Y  ->  ( U  X.  { w }
)  =  ( U  X.  { Y }
) )
6564reseq2d 4971 . . . . . . . . . . 11  |-  ( w  =  Y  ->  ( K  |`  ( U  X.  { w } ) )  =  ( K  |`  ( U  X.  { Y } ) ) )
6664oveq2d 5890 . . . . . . . . . . . 12  |-  ( w  =  Y  ->  (
( II  tX  II )t  ( U  X.  { w } ) )  =  ( ( II  tX  II )t  ( U  X.  { Y } ) ) )
6766oveq1d 5889 . . . . . . . . . . 11  |-  ( w  =  Y  ->  (
( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C )  =  ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  Cn  C
) )
6865, 67eleq12d 2364 . . . . . . . . . 10  |-  ( w  =  Y  ->  (
( K  |`  ( U  X.  { w }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C )  <-> 
( K  |`  ( U  X.  { Y }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  Cn  C ) ) )
6968rspcev 2897 . . . . . . . . 9  |-  ( ( Y  e.  V  /\  ( K  |`  ( U  X.  { Y }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  Cn  C ) )  ->  E. w  e.  V  ( K  |`  ( U  X.  { w }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C ) )
7017, 62, 69syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  E. w  e.  V  ( K  |`  ( U  X.  { w }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C ) )
71 cvmlift2lem11.5 . . . . . . . . 9  |-  ( ph  ->  ( E. w  e.  V  ( K  |`  ( U  X.  { w } ) )  e.  ( ( ( II 
tX  II )t  ( U  X.  { w }
) )  Cn  C
)  ->  ( K  |`  ( U  X.  V
) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C ) ) )
7271imp 418 . . . . . . . 8  |-  ( (
ph  /\  E. w  e.  V  ( K  |`  ( U  X.  {
w } ) )  e.  ( ( ( II  tX  II )t  ( U  X.  { w }
) )  Cn  C
) )  ->  ( K  |`  ( U  X.  V ) )  e.  ( ( ( II 
tX  II )t  ( U  X.  V ) )  Cn  C ) )
7370, 72syldan 456 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  V ) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C ) )
7473adantr 451 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  ( K  |`  ( U  X.  V ) )  e.  ( ( ( II 
tX  II )t  ( U  X.  V ) )  Cn  C ) )
757adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Z  e.  V )
7675snssd 3776 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Z }  C_  V
)
77 xpss2 4812 . . . . . . . . 9  |-  ( { Z }  C_  V  ->  ( U  X.  { Z } )  C_  ( U  X.  V ) )
7876, 77syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  ( U  X.  V ) )
79 iitop 18400 . . . . . . . . . 10  |-  II  e.  Top
8079, 79txtopi 17301 . . . . . . . . 9  |-  ( II 
tX  II )  e. 
Top
81 xpss12 4808 . . . . . . . . . 10  |-  ( ( U  C_  ( 0 [,] 1 )  /\  V  C_  ( 0 [,] 1 ) )  -> 
( U  X.  V
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
826, 30, 81syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
8349restuni 16909 . . . . . . . . 9  |-  ( ( ( II  tX  II )  e.  Top  /\  ( U  X.  V )  C_  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  ->  ( U  X.  V )  = 
U. ( ( II 
tX  II )t  ( U  X.  V ) ) )
8480, 82, 83sylancr 644 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  =  U. (
( II  tX  II )t  ( U  X.  V
) ) )
8578, 84sseqtrd 3227 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  U. (
( II  tX  II )t  ( U  X.  V
) ) )
8685sselda 3193 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  z  e.  U. ( ( II 
tX  II )t  ( U  X.  V ) ) )
87 eqid 2296 . . . . . . 7  |-  U. (
( II  tX  II )t  ( U  X.  V
) )  =  U. ( ( II  tX  II )t  ( U  X.  V ) )
8887cncnpi 17023 . . . . . 6  |-  ( ( ( K  |`  ( U  X.  V ) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C )  /\  z  e.  U. (
( II  tX  II )t  ( U  X.  V
) ) )  -> 
( K  |`  ( U  X.  V ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  V
) )  CnP  C
) `  z )
)
8974, 86, 88syl2anc 642 . . . . 5  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  ( K  |`  ( U  X.  V ) )  e.  ( ( ( ( II  tX  II )t  ( U  X.  V ) )  CnP  C ) `  z ) )
9080a1i 10 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  (
II  tX  II )  e.  Top )
9182adantr 451 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  ( U  X.  V )  C_  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
9279a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  II  e.  Top )
93 txopn 17313 . . . . . . . . . 10  |-  ( ( ( II  e.  Top  /\  II  e.  Top )  /\  ( U  e.  II  /\  V  e.  II ) )  ->  ( U  X.  V )  e.  ( II  tX  II ) )
9492, 92, 2, 27, 93syl22anc 1183 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  e.  ( II 
tX  II ) )
95 isopn3i 16835 . . . . . . . . 9  |-  ( ( ( II  tX  II )  e.  Top  /\  ( U  X.  V )  e.  ( II  tX  II ) )  ->  (
( int `  (
II  tX  II )
) `  ( U  X.  V ) )  =  ( U  X.  V
) )
9680, 94, 95sylancr 644 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( ( int `  (
II  tX  II )
) `  ( U  X.  V ) )  =  ( U  X.  V
) )
9778, 96sseqtr4d 3228 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  (
( int `  (
II  tX  II )
) `  ( U  X.  V ) ) )
9897sselda 3193 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  z  e.  ( ( int `  (
II  tX  II )
) `  ( U  X.  V ) ) )
9925ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
10049, 18cnprest 17033 . . . . . 6  |-  ( ( ( ( II  tX  II )  e.  Top  /\  ( U  X.  V
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  /\  ( z  e.  ( ( int `  (
II  tX  II )
) `  ( U  X.  V ) )  /\  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B ) )  ->  ( K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
)  <->  ( K  |`  ( U  X.  V
) )  e.  ( ( ( ( II 
tX  II )t  ( U  X.  V ) )  CnP  C ) `  z ) ) )
10190, 91, 98, 99, 100syl22anc 1183 . . . . 5  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  ( K  e.  ( (
( II  tX  II )  CnP  C ) `  z )  <->  ( K  |`  ( U  X.  V
) )  e.  ( ( ( ( II 
tX  II )t  ( U  X.  V ) )  CnP  C ) `  z ) ) )
10289, 101mpbird 223 . . . 4  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
10315, 102ssrabdv 3265 . . 3  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  { z  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  |  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) } )
104103, 40syl6sseqr 3238 . 2  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  M
)
105104ex 423 1  |-  ( ph  ->  ( ( U  X.  { Y } )  C_  M  ->  ( U  X.  { Z } )  C_  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   {csn 3653   U.cuni 3843    e. cmpt 4093    X. cxp 4703    |` cres 4707    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   iota_crio 6313   0cc0 8753   1c1 8754   [,]cicc 10675   ↾t crest 13341   Topctop 16647  TopOnctopon 16648   intcnt 16770    Cn ccn 16970    CnP ccnp 16971    tX ctx 17271   IIcii 18395   CovMap ccvm 23801
This theorem is referenced by:  cvmlift2lem12  23860
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  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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-nel 2462  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-iin 3924  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-cn 16973  df-cnp 16974  df-cmp 17130  df-con 17154  df-lly 17208  df-nlly 17209  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-ii 18397  df-htpy 18484  df-phtpy 18485  df-phtpc 18506  df-pcon 23767  df-scon 23768  df-cvm 23802
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