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Theorem cvmlift2lem11 27207
Description: Lemma for cvmlift2 27210. (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 465 . . . . . 6  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  U  e.  II )
3 elssuni 4126 . . . . . . 7  |-  ( U  e.  II  ->  U  C_ 
U. II )
4 iiuni 20462 . . . . . . 7  |-  ( 0 [,] 1 )  = 
U. II
53, 4syl6sseqr 3408 . . . . . 6  |-  ( U  e.  II  ->  U  C_  ( 0 [,] 1
) )
62, 5syl 16 . . . . 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 4101 . . . . . . . . 9  |-  ( ( Z  e.  V  /\  V  e.  II )  ->  Z  e.  U. II )
109, 4syl6eleqr 2534 . . . . . . . 8  |-  ( ( Z  e.  V  /\  V  e.  II )  ->  Z  e.  ( 0 [,] 1 ) )
117, 8, 10syl2anc 661 . . . . . . 7  |-  ( ph  ->  Z  e.  ( 0 [,] 1 ) )
1211adantr 465 . . . . . 6  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Z  e.  ( 0 [,] 1 ) )
1312snssd 4023 . . . . 5  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Z }  C_  (
0 [,] 1 ) )
14 xpss12 4950 . . . . 5  |-  ( ( U  C_  ( 0 [,] 1 )  /\  { Z }  C_  (
0 [,] 1 ) )  ->  ( U  X.  { Z } ) 
C_  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
156, 13, 14syl2anc 661 . . . 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 465 . . . . . . . . 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 27201 . . . . . . . . . . . 12  |-  ( ph  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B )
2625adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
278adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  V  e.  II )
28 elssuni 4126 . . . . . . . . . . . . . . . 16  |-  ( V  e.  II  ->  V  C_ 
U. II )
2928, 4syl6sseqr 3408 . . . . . . . . . . . . . . 15  |-  ( V  e.  II  ->  V  C_  ( 0 [,] 1
) )
3027, 29syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  V  C_  ( 0 [,] 1 ) )
3130, 17sseldd 3362 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Y  e.  ( 0 [,] 1 ) )
3231snssd 4023 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Y }  C_  (
0 [,] 1 ) )
33 xpss12 4950 . . . . . . . . . . . 12  |-  ( ( U  C_  ( 0 [,] 1 )  /\  { Y }  C_  (
0 [,] 1 ) )  ->  ( U  X.  { Y } ) 
C_  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
346, 32, 33syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Y } )  C_  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
35 fssres 5583 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  { Y }
) ) : ( U  X.  { Y } ) --> B )
3734adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  ( U  X.  { Y }
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
38 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  z  e.  ( U  X.  { Y } ) )
39 simpr 461 . . . . . . . . . . . . . . 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 3407 . . . . . . . . . . . . . 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 3435 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  A. z  e.  ( U  X.  { Y }
) K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) )
4544r19.21bi 2819 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
46 iitopon 20460 . . . . . . . . . . . . . . 15  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
47 txtopon 19169 . . . . . . . . . . . . . . 15  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  II  e.  (TopOn `  ( 0 [,] 1 ) ) )  ->  ( II  tX  II )  e.  (TopOn `  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
4846, 46, 47mp2an 672 . . . . . . . . . . . . . 14  |-  ( II 
tX  II )  e.  (TopOn `  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
4948toponunii 18542 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
5049cnpresti 18897 . . . . . . . . . . . 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 1218 . . . . . . . . . . 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 2804 . . . . . . . . . 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 18770 . . . . . . . . . . . 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 663 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( ( II  tX  II )t  ( U  X.  { Y } ) )  e.  (TopOn `  ( U  X.  { Y }
) ) )
55 cvmtop1 27154 . . . . . . . . . . . . . 14  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
5619, 55syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  Top )
5756adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  C  e.  Top )
5818toptopon 18543 . . . . . . . . . . . 12  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
5957, 58sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  C  e.  (TopOn `  B
) )
60 cncnp 18889 . . . . . . . . . . 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 661 . . . . . . . . . 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 913 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  { Y }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  Cn  C ) )
63 sneq 3892 . . . . . . . . . . . . 13  |-  ( w  =  Y  ->  { w }  =  { Y } )
6463xpeq2d 4869 . . . . . . . . . . . 12  |-  ( w  =  Y  ->  ( U  X.  { w }
)  =  ( U  X.  { Y }
) )
6564reseq2d 5115 . . . . . . . . . . 11  |-  ( w  =  Y  ->  ( K  |`  ( U  X.  { w } ) )  =  ( K  |`  ( U  X.  { Y } ) ) )
6664oveq2d 6112 . . . . . . . . . . . 12  |-  ( w  =  Y  ->  (
( II  tX  II )t  ( U  X.  { w } ) )  =  ( ( II  tX  II )t  ( U  X.  { Y } ) ) )
6766oveq1d 6111 . . . . . . . . . . 11  |-  ( w  =  Y  ->  (
( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C )  =  ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  Cn  C
) )
6865, 67eleq12d 2511 . . . . . . . . . 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 3078 . . . . . . . . 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 661 . . . . . . . 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 429 . . . . . . . 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 470 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  V ) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C ) )
7473adantr 465 . . . . . 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 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Z  e.  V )
7675snssd 4023 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Z }  C_  V
)
77 xpss2 4954 . . . . . . . . 9  |-  ( { Z }  C_  V  ->  ( U  X.  { Z } )  C_  ( U  X.  V ) )
7876, 77syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  ( U  X.  V ) )
79 iitop 20461 . . . . . . . . . 10  |-  II  e.  Top
8079, 79txtopi 19168 . . . . . . . . 9  |-  ( II 
tX  II )  e. 
Top
81 xpss12 4950 . . . . . . . . . 10  |-  ( ( U  C_  ( 0 [,] 1 )  /\  V  C_  ( 0 [,] 1 ) )  -> 
( U  X.  V
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
826, 30, 81syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
8349restuni 18771 . . . . . . . . 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 663 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  =  U. (
( II  tX  II )t  ( U  X.  V
) ) )
8578, 84sseqtrd 3397 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  U. (
( II  tX  II )t  ( U  X.  V
) ) )
8685sselda 3361 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  z  e.  U. ( ( II 
tX  II )t  ( U  X.  V ) ) )
87 eqid 2443 . . . . . . 7  |-  U. (
( II  tX  II )t  ( U  X.  V
) )  =  U. ( ( II  tX  II )t  ( U  X.  V ) )
8887cncnpi 18887 . . . . . 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 661 . . . . 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 11 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  (
II  tX  II )  e.  Top )
9182adantr 465 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  ( U  X.  V )  C_  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
9279a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  II  e.  Top )
93 txopn 19180 . . . . . . . . . 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 1219 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  e.  ( II 
tX  II ) )
95 isopn3i 18691 . . . . . . . . 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 663 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( ( int `  (
II  tX  II )
) `  ( U  X.  V ) )  =  ( U  X.  V
) )
9778, 96sseqtr4d 3398 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  (
( int `  (
II  tX  II )
) `  ( U  X.  V ) ) )
9897sselda 3361 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  z  e.  ( ( int `  (
II  tX  II )
) `  ( U  X.  V ) ) )
9925ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
10049, 18cnprest 18898 . . . . . 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 1219 . . . . 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 232 . . . 4  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
10315, 102ssrabdv 3436 . . 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 3408 . 2  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  M
)
105104ex 434 1  |-  ( ph  ->  ( ( U  X.  { Y } )  C_  M  ->  ( U  X.  { Z } )  C_  M ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   {crab 2724    C_ wss 3333   {csn 3882   U.cuni 4096    e. cmpt 4355    X. cxp 4843    |` cres 4847    o. ccom 4849   -->wf 5419   ` cfv 5423   iota_crio 6056  (class class class)co 6096    e. cmpt2 6098   0cc0 9287   1c1 9288   [,]cicc 11308   ↾t crest 14364   Topctop 18503  TopOnctopon 18504   intcnt 18626    Cn ccn 18833    CnP ccnp 18834    tX ctx 19138   IIcii 20456   CovMap ccvm 27149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-ec 7108  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-cn 18836  df-cnp 18837  df-cmp 18995  df-con 19021  df-lly 19075  df-nlly 19076  df-tx 19140  df-hmeo 19333  df-xms 19900  df-ms 19901  df-tms 19902  df-ii 20458  df-htpy 20547  df-phtpy 20548  df-phtpc 20569  df-pcon 27115  df-scon 27116  df-cvm 27150
This theorem is referenced by:  cvmlift2lem12  27208
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