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Theorem cvmlift2lem11 29486
Description: Lemma for cvmlift2 29489. (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 463 . . . . . 6  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  U  e.  II )
3 elssuni 4217 . . . . . . 7  |-  ( U  e.  II  ->  U  C_ 
U. II )
4 iiuni 21567 . . . . . . 7  |-  ( 0 [,] 1 )  = 
U. II
53, 4syl6sseqr 3486 . . . . . 6  |-  ( U  e.  II  ->  U  C_  ( 0 [,] 1
) )
62, 5syl 17 . . . . 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 4193 . . . . . . . . 9  |-  ( ( Z  e.  V  /\  V  e.  II )  ->  Z  e.  U. II )
109, 4syl6eleqr 2499 . . . . . . . 8  |-  ( ( Z  e.  V  /\  V  e.  II )  ->  Z  e.  ( 0 [,] 1 ) )
117, 8, 10syl2anc 659 . . . . . . 7  |-  ( ph  ->  Z  e.  ( 0 [,] 1 ) )
1211adantr 463 . . . . . 6  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Z  e.  ( 0 [,] 1 ) )
1312snssd 4114 . . . . 5  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Z }  C_  (
0 [,] 1 ) )
14 xpss12 5048 . . . . 5  |-  ( ( U  C_  ( 0 [,] 1 )  /\  { Z }  C_  (
0 [,] 1 ) )  ->  ( U  X.  { Z } ) 
C_  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
156, 13, 14syl2anc 659 . . . 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 463 . . . . . . . . 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 29480 . . . . . . . . . . . 12  |-  ( ph  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B )
2625adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
278adantr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  V  e.  II )
28 elssuni 4217 . . . . . . . . . . . . . . . 16  |-  ( V  e.  II  ->  V  C_ 
U. II )
2928, 4syl6sseqr 3486 . . . . . . . . . . . . . . 15  |-  ( V  e.  II  ->  V  C_  ( 0 [,] 1
) )
3027, 29syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  V  C_  ( 0 [,] 1 ) )
3130, 17sseldd 3440 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Y  e.  ( 0 [,] 1 ) )
3231snssd 4114 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Y }  C_  (
0 [,] 1 ) )
33 xpss12 5048 . . . . . . . . . . . 12  |-  ( ( U  C_  ( 0 [,] 1 )  /\  { Y }  C_  (
0 [,] 1 ) )  ->  ( U  X.  { Y } ) 
C_  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
346, 32, 33syl2anc 659 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Y } )  C_  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
3526, 34fssresd 5689 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  { Y }
) ) : ( U  X.  { Y } ) --> B )
3634adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  ( U  X.  { Y }
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
37 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  z  e.  ( U  X.  { Y } ) )
38 simpr 459 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Y } )  C_  M
)
39 cvmlift2.m . . . . . . . . . . . . . . 15  |-  M  =  { z  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) )  |  K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) }
4038, 39syl6sseq 3485 . . . . . . . . . . . . . 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
) } )
41 ssrab 3514 . . . . . . . . . . . . . . 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 ) ) )
4241simprbi 462 . . . . . . . . . . . . . 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
) )
4340, 42syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  A. z  e.  ( U  X.  { Y }
) K  e.  ( ( ( II  tX  II )  CnP  C ) `
 z ) )
4443r19.21bi 2770 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Y }
) )  ->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
45 iitopon 21565 . . . . . . . . . . . . . . 15  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
46 txtopon 20274 . . . . . . . . . . . . . . 15  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  II  e.  (TopOn `  ( 0 [,] 1 ) ) )  ->  ( II  tX  II )  e.  (TopOn `  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
4745, 45, 46mp2an 670 . . . . . . . . . . . . . 14  |-  ( II 
tX  II )  e.  (TopOn `  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
4847toponunii 19615 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
4948cnpresti 19972 . . . . . . . . . . . 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
) )
5036, 37, 44, 49syl3anc 1228 . . . . . . . . . . 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
) )
5150ralrimiva 2815 . . . . . . . . . 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 )
)
52 resttopon 19845 . . . . . . . . . . . 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 } ) ) )
5347, 34, 52sylancr 661 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( ( II  tX  II )t  ( U  X.  { Y } ) )  e.  (TopOn `  ( U  X.  { Y }
) ) )
54 cvmtop1 29433 . . . . . . . . . . . . . 14  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
5519, 54syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  Top )
5655adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  C  e.  Top )
5718toptopon 19616 . . . . . . . . . . . 12  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
5856, 57sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  C  e.  (TopOn `  B
) )
59 cncnp 19964 . . . . . . . . . . 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 )
) ) )
6053, 58, 59syl2anc 659 . . . . . . . . . 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 )
) ) )
6135, 51, 60mpbir2and 921 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  { Y }
) )  e.  ( ( ( II  tX  II )t  ( U  X.  { Y } ) )  Cn  C ) )
62 sneq 3979 . . . . . . . . . . . . 13  |-  ( w  =  Y  ->  { w }  =  { Y } )
6362xpeq2d 4964 . . . . . . . . . . . 12  |-  ( w  =  Y  ->  ( U  X.  { w }
)  =  ( U  X.  { Y }
) )
6463reseq2d 5213 . . . . . . . . . . 11  |-  ( w  =  Y  ->  ( K  |`  ( U  X.  { w } ) )  =  ( K  |`  ( U  X.  { Y } ) ) )
6563oveq2d 6248 . . . . . . . . . . . 12  |-  ( w  =  Y  ->  (
( II  tX  II )t  ( U  X.  { w } ) )  =  ( ( II  tX  II )t  ( U  X.  { Y } ) ) )
6665oveq1d 6247 . . . . . . . . . . 11  |-  ( w  =  Y  ->  (
( ( II  tX  II )t  ( U  X.  { w } ) )  Cn  C )  =  ( ( ( II  tX  II )t  ( U  X.  { Y }
) )  Cn  C
) )
6764, 66eleq12d 2482 . . . . . . . . . 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 ) ) )
6867rspcev 3157 . . . . . . . . 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 ) )
6917, 61, 68syl2anc 659 . . . . . . . 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 ) )
70 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 ) ) )
7170imp 427 . . . . . . . 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 ) )
7269, 71syldan 468 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( K  |`  ( U  X.  V ) )  e.  ( ( ( II  tX  II )t  ( U  X.  V ) )  Cn  C ) )
7372adantr 463 . . . . . 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 ) )
747adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  Z  e.  V )
7574snssd 4114 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  { Z }  C_  V
)
76 xpss2 5052 . . . . . . . . 9  |-  ( { Z }  C_  V  ->  ( U  X.  { Z } )  C_  ( U  X.  V ) )
7775, 76syl 17 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  ( U  X.  V ) )
78 iitop 21566 . . . . . . . . . 10  |-  II  e.  Top
7978, 78txtopi 20273 . . . . . . . . 9  |-  ( II 
tX  II )  e. 
Top
80 xpss12 5048 . . . . . . . . . 10  |-  ( ( U  C_  ( 0 [,] 1 )  /\  V  C_  ( 0 [,] 1 ) )  -> 
( U  X.  V
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
816, 30, 80syl2anc 659 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  C_  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
8248restuni 19846 . . . . . . . . 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 ) ) )
8379, 81, 82sylancr 661 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  =  U. (
( II  tX  II )t  ( U  X.  V
) ) )
8477, 83sseqtrd 3475 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  U. (
( II  tX  II )t  ( U  X.  V
) ) )
8584sselda 3439 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  z  e.  U. ( ( II 
tX  II )t  ( U  X.  V ) ) )
86 eqid 2400 . . . . . . 7  |-  U. (
( II  tX  II )t  ( U  X.  V
) )  =  U. ( ( II  tX  II )t  ( U  X.  V ) )
8786cncnpi 19962 . . . . . 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 )
)
8873, 85, 87syl2anc 659 . . . . 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 ) )
8979a1i 11 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  (
II  tX  II )  e.  Top )
9081adantr 463 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  ( U  X.  V )  C_  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
9178a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  ->  II  e.  Top )
92 txopn 20285 . . . . . . . . . 10  |-  ( ( ( II  e.  Top  /\  II  e.  Top )  /\  ( U  e.  II  /\  V  e.  II ) )  ->  ( U  X.  V )  e.  ( II  tX  II ) )
9391, 91, 2, 27, 92syl22anc 1229 . . . . . . . . 9  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  V
)  e.  ( II 
tX  II ) )
94 isopn3i 19766 . . . . . . . . 9  |-  ( ( ( II  tX  II )  e.  Top  /\  ( U  X.  V )  e.  ( II  tX  II ) )  ->  (
( int `  (
II  tX  II )
) `  ( U  X.  V ) )  =  ( U  X.  V
) )
9579, 93, 94sylancr 661 . . . . . . . 8  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( ( int `  (
II  tX  II )
) `  ( U  X.  V ) )  =  ( U  X.  V
) )
9677, 95sseqtr4d 3476 . . . . . . 7  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  (
( int `  (
II  tX  II )
) `  ( U  X.  V ) ) )
9796sselda 3439 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  z  e.  ( ( int `  (
II  tX  II )
) `  ( U  X.  V ) ) )
9825ad2antrr 724 . . . . . 6  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
9948, 18cnprest 19973 . . . . . 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 ) ) )
10089, 90, 97, 98, 99syl22anc 1229 . . . . 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 ) ) )
10188, 100mpbird 232 . . . 4  |-  ( ( ( ph  /\  ( U  X.  { Y }
)  C_  M )  /\  z  e.  ( U  X.  { Z }
) )  ->  K  e.  ( ( ( II 
tX  II )  CnP 
C ) `  z
) )
10215, 101ssrabdv 3515 . . 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
) } )
103102, 39syl6sseqr 3486 . 2  |-  ( (
ph  /\  ( U  X.  { Y } ) 
C_  M )  -> 
( U  X.  { Z } )  C_  M
)
104103ex 432 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 367    = wceq 1403    e. wcel 1840   A.wral 2751   E.wrex 2752   {crab 2755    C_ wss 3411   {csn 3969   U.cuni 4188    |-> cmpt 4450    X. cxp 4938    |` cres 4942    o. ccom 4944   -->wf 5519   ` cfv 5523   iota_crio 6193  (class class class)co 6232    |-> cmpt2 6234   0cc0 9440   1c1 9441   [,]cicc 11501   ↾t crest 14925   Topctop 19576  TopOnctopon 19577   intcnt 19700    Cn ccn 19908    CnP ccnp 19909    tX ctx 20243   IIcii 21561   CovMap ccvm 29428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-ec 7268  df-map 7377  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-fi 7823  df-sup 7853  df-oi 7887  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-q 11144  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-ioo 11502  df-ico 11504  df-icc 11505  df-fz 11642  df-fzo 11766  df-fl 11877  df-seq 12060  df-exp 12119  df-hash 12358  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-clim 13365  df-sum 13563  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-hom 14823  df-cco 14824  df-rest 14927  df-topn 14928  df-0g 14946  df-gsum 14947  df-topgen 14948  df-pt 14949  df-prds 14952  df-xrs 15006  df-qtop 15011  df-imas 15012  df-xps 15014  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-submnd 16181  df-mulg 16274  df-cntz 16569  df-cmn 17014  df-psmet 18621  df-xmet 18622  df-met 18623  df-bl 18624  df-mopn 18625  df-cnfld 18631  df-top 19581  df-bases 19583  df-topon 19584  df-topsp 19585  df-cld 19702  df-ntr 19703  df-cls 19704  df-nei 19782  df-cn 19911  df-cnp 19912  df-cmp 20070  df-con 20095  df-lly 20149  df-nlly 20150  df-tx 20245  df-hmeo 20438  df-xms 21005  df-ms 21006  df-tms 21007  df-ii 21563  df-htpy 21652  df-phtpy 21653  df-phtpc 21674  df-pcon 29394  df-scon 29395  df-cvm 29429
This theorem is referenced by:  cvmlift2lem12  29487
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