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Theorem cvmlift3lem9 29624
Description: Lemma for cvmlift2 29613. (Contributed by Mario Carneiro, 7-May-2015.)
Hypotheses
Ref Expression
cvmlift3.b  |-  B  = 
U. C
cvmlift3.y  |-  Y  = 
U. K
cvmlift3.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmlift3.k  |-  ( ph  ->  K  e. SCon )
cvmlift3.l  |-  ( ph  ->  K  e. 𝑛Locally PCon )
cvmlift3.o  |-  ( ph  ->  O  e.  Y )
cvmlift3.g  |-  ( ph  ->  G  e.  ( K  Cn  J ) )
cvmlift3.p  |-  ( ph  ->  P  e.  B )
cvmlift3.e  |-  ( ph  ->  ( F `  P
)  =  ( G `
 O ) )
cvmlift3.h  |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B  E. f  e.  (
II  Cn  K )
( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  z ) ) )
cvmlift3lem7.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 ) ) ) ) } )
Assertion
Ref Expression
cvmlift3lem9  |-  ( ph  ->  E. f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
Distinct variable groups:    c, d,
f, k, s, z, g, x    J, c   
g, d, x, J, f, k, s    F, c, d, f, g, k, s    x, z, F    H, c, d, f, g, x, z    S, f, x    B, d, f, g, x, z    G, c, d, f, g, k, x, z    C, c, d, f, g, k, s, x, z    ph, f, x    K, c, f, g, x, z    P, c, d, f, g, x, z    O, c, f, g, x, z    f, Y, g, x, z
Allowed substitution hints:    ph( z, g, k, s, c, d)    B( k, s, c)    P( k, s)    S( z, g, k, s, c, d)    G( s)    H( k, s)    J( z)    K( k, s, d)    O( k, s, d)    Y( k, s, c, d)

Proof of Theorem cvmlift3lem9
StepHypRef Expression
1 cvmlift3.b . . 3  |-  B  = 
U. C
2 cvmlift3.y . . 3  |-  Y  = 
U. K
3 cvmlift3.f . . 3  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
4 cvmlift3.k . . 3  |-  ( ph  ->  K  e. SCon )
5 cvmlift3.l . . 3  |-  ( ph  ->  K  e. 𝑛Locally PCon )
6 cvmlift3.o . . 3  |-  ( ph  ->  O  e.  Y )
7 cvmlift3.g . . 3  |-  ( ph  ->  G  e.  ( K  Cn  J ) )
8 cvmlift3.p . . 3  |-  ( ph  ->  P  e.  B )
9 cvmlift3.e . . 3  |-  ( ph  ->  ( F `  P
)  =  ( G `
 O ) )
10 cvmlift3.h . . 3  |-  H  =  ( x  e.  Y  |->  ( iota_ z  e.  B  E. f  e.  (
II  Cn  K )
( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  x  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  z ) ) )
11 cvmlift3lem7.s . . 3  |-  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 ) ) ) ) } )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem8 29623 . 2  |-  ( ph  ->  H  e.  ( K  Cn  C ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem5 29620 . 2  |-  ( ph  ->  ( F  o.  H
)  =  G )
14 iitopon 21675 . . . . . 6  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
1514a1i 11 . . . . 5  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
16 scontop 29525 . . . . . . 7  |-  ( K  e. SCon  ->  K  e.  Top )
174, 16syl 17 . . . . . 6  |-  ( ph  ->  K  e.  Top )
182toptopon 19726 . . . . . 6  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
1917, 18sylib 196 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
20 cnconst2 20077 . . . . 5  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  K  e.  (TopOn `  Y )  /\  O  e.  Y
)  ->  ( (
0 [,] 1 )  X.  { O }
)  e.  ( II 
Cn  K ) )
2115, 19, 6, 20syl3anc 1230 . . . 4  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  { O } )  e.  ( II  Cn  K ) )
22 0elunit 11692 . . . . 5  |-  0  e.  ( 0 [,] 1
)
23 fvconst2g 6105 . . . . 5  |-  ( ( O  e.  Y  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O )
246, 22, 23sylancl 660 . . . 4  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 0 )  =  O )
25 1elunit 11693 . . . . 5  |-  1  e.  ( 0 [,] 1
)
26 fvconst2g 6105 . . . . 5  |-  ( ( O  e.  Y  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { O }
) `  1 )  =  O )
276, 25, 26sylancl 660 . . . 4  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 1 )  =  O )
289sneqd 3984 . . . . . . . . 9  |-  ( ph  ->  { ( F `  P ) }  =  { ( G `  O ) } )
2928xpeq2d 4847 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( F `  P
) } )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
30 cvmcn 29559 . . . . . . . . . 10  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
31 eqid 2402 . . . . . . . . . . 11  |-  U. J  =  U. J
321, 31cnf 20040 . . . . . . . . . 10  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
33 ffn 5714 . . . . . . . . . 10  |-  ( F : B --> U. J  ->  F  Fn  B )
343, 30, 32, 334syl 19 . . . . . . . . 9  |-  ( ph  ->  F  Fn  B )
35 fcoconst 6047 . . . . . . . . 9  |-  ( ( F  Fn  B  /\  P  e.  B )  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  P
) } ) )
3634, 8, 35syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  P
) } ) )
372, 31cnf 20040 . . . . . . . . . . 11  |-  ( G  e.  ( K  Cn  J )  ->  G : Y --> U. J )
387, 37syl 17 . . . . . . . . . 10  |-  ( ph  ->  G : Y --> U. J
)
39 ffn 5714 . . . . . . . . . 10  |-  ( G : Y --> U. J  ->  G  Fn  Y )
4038, 39syl 17 . . . . . . . . 9  |-  ( ph  ->  G  Fn  Y )
41 fcoconst 6047 . . . . . . . . 9  |-  ( ( G  Fn  Y  /\  O  e.  Y )  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
4240, 6, 41syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
4329, 36, 423eqtr4d 2453 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) ) )
44 fvconst2g 6105 . . . . . . . 8  |-  ( ( P  e.  B  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { P }
) `  0 )  =  P )
458, 22, 44sylancl 660 . . . . . . 7  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 0 )  =  P )
46 cvmtop1 29557 . . . . . . . . . . 11  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
473, 46syl 17 . . . . . . . . . 10  |-  ( ph  ->  C  e.  Top )
481toptopon 19726 . . . . . . . . . 10  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
4947, 48sylib 196 . . . . . . . . 9  |-  ( ph  ->  C  e.  (TopOn `  B ) )
50 cnconst2 20077 . . . . . . . . 9  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  C  e.  (TopOn `  B )  /\  P  e.  B
)  ->  ( (
0 [,] 1 )  X.  { P }
)  e.  ( II 
Cn  C ) )
5115, 49, 8, 50syl3anc 1230 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  { P } )  e.  ( II  Cn  C ) )
52 cvmtop2 29558 . . . . . . . . . . . . 13  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
533, 52syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  Top )
5431toptopon 19726 . . . . . . . . . . . 12  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
5553, 54sylib 196 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
5638, 6ffvelrnd 6010 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  O
)  e.  U. J
)
57 cnconst2 20077 . . . . . . . . . . 11  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  J  e.  (TopOn `  U. J )  /\  ( G `  O )  e.  U. J )  ->  (
( 0 [,] 1
)  X.  { ( G `  O ) } )  e.  ( II  Cn  J ) )
5815, 55, 56, 57syl3anc 1230 . . . . . . . . . 10  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( G `  O
) } )  e.  ( II  Cn  J
) )
5942, 58eqeltrd 2490 . . . . . . . . 9  |-  ( ph  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  e.  ( II  Cn  J
) )
60 fvconst2g 6105 . . . . . . . . . . 11  |-  ( ( ( G `  O
)  e.  U. J  /\  0  e.  (
0 [,] 1 ) )  ->  ( (
( 0 [,] 1
)  X.  { ( G `  O ) } ) `  0
)  =  ( G `
 O ) )
6156, 22, 60sylancl 660 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ ( G `  O ) } ) `
 0 )  =  ( G `  O
) )
6242fveq1d 5851 . . . . . . . . . 10  |-  ( ph  ->  ( ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) ) ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) ` 
0 ) )
6361, 62, 93eqtr4rd 2454 . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  =  ( ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) `  0
) )
641cvmlift 29596 . . . . . . . . 9  |-  ( ( ( F  e.  ( C CovMap  J )  /\  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  e.  ( II  Cn  J
) )  /\  ( P  e.  B  /\  ( F `  P )  =  ( ( G  o.  ( ( 0 [,] 1 )  X. 
{ O } ) ) `  0 ) ) )  ->  E! g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) )  /\  ( g `  0
)  =  P ) )
653, 59, 8, 63, 64syl22anc 1231 . . . . . . . 8  |-  ( ph  ->  E! g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )
66 coeq2 4982 . . . . . . . . . . 11  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( F  o.  g )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { P }
) ) )
6766eqeq1d 2404 . . . . . . . . . 10  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  <->  ( F  o.  ( ( 0 [,] 1 )  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) ) ) )
68 fveq1 5848 . . . . . . . . . . 11  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( g ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  { P } ) `  0
) )
6968eqeq1d 2404 . . . . . . . . . 10  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( ( g `
 0 )  =  P  <->  ( ( ( 0 [,] 1 )  X.  { P }
) `  0 )  =  P ) )
7067, 69anbi12d 709 . . . . . . . . 9  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) )  /\  ( g `  0
)  =  P )  <-> 
( ( F  o.  ( ( 0 [,] 1 )  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 0 )  =  P ) ) )
7170riota2 6262 . . . . . . . 8  |-  ( ( ( ( 0 [,] 1 )  X.  { P } )  e.  ( II  Cn  C )  /\  E! g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) )  ->  ( (
( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 0 )  =  P )  <->  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) )  /\  ( g `  0
)  =  P ) )  =  ( ( 0 [,] 1 )  X.  { P }
) ) )
7251, 65, 71syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( ( ( F  o.  ( ( 0 [,] 1 )  X. 
{ P } ) )  =  ( G  o.  ( ( 0 [,] 1 )  X. 
{ O } ) )  /\  ( ( ( 0 [,] 1
)  X.  { P } ) `  0
)  =  P )  <-> 
( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )  =  ( ( 0 [,] 1 )  X.  { P } ) ) )
7343, 45, 72mpbi2and 922 . . . . . 6  |-  ( ph  ->  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )  =  ( ( 0 [,] 1 )  X.  { P } ) )
7473fveq1d 5851 . . . . 5  |-  ( ph  ->  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) `  1 )  =  ( ( ( 0 [,] 1 )  X.  { P }
) `  1 )
)
75 fvconst2g 6105 . . . . . 6  |-  ( ( P  e.  B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { P }
) `  1 )  =  P )
768, 25, 75sylancl 660 . . . . 5  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 1 )  =  P )
7774, 76eqtrd 2443 . . . 4  |-  ( ph  ->  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) `  1 )  =  P )
78 fveq1 5848 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( f ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  { O } ) `  0
) )
7978eqeq1d 2404 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( f `
 0 )  =  O  <->  ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O ) )
80 fveq1 5848 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( f ` 
1 )  =  ( ( ( 0 [,] 1 )  X.  { O } ) `  1
) )
8180eqeq1d 2404 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( f `
 1 )  =  O  <->  ( ( ( 0 [,] 1 )  X.  { O }
) `  1 )  =  O ) )
82 coeq2 4982 . . . . . . . . . . 11  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( G  o.  f )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) )
8382eqeq2d 2416 . . . . . . . . . 10  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( F  o.  g )  =  ( G  o.  f
)  <->  ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) ) )
8483anbi1d 703 . . . . . . . . 9  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P )  <->  ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) )
8584riotabidv 6242 . . . . . . . 8  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  f
)  /\  ( g `  0 )  =  P ) )  =  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) ) )
8685fveq1d 5851 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) ) ` 
1 ) )
8786eqeq1d 2404 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  P  <->  ( ( iota_ g  e.  ( II 
Cn  C ) ( ( F  o.  g
)  =  ( G  o.  ( ( 0 [,] 1 )  X. 
{ O } ) )  /\  ( g `
 0 )  =  P ) ) ` 
1 )  =  P ) )
8879, 81, 873anbi123d 1301 . . . . 5  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( ( f `  0 )  =  O  /\  (
f `  1 )  =  O  /\  (
( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  P )  <-> 
( ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O  /\  (
( ( 0 [,] 1 )  X.  { O } ) `  1
)  =  O  /\  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) `  1 )  =  P ) ) )
8988rspcev 3160 . . . 4  |-  ( ( ( ( 0 [,] 1 )  X.  { O } )  e.  ( II  Cn  K )  /\  ( ( ( ( 0 [,] 1
)  X.  { O } ) `  0
)  =  O  /\  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 1 )  =  O  /\  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  P ) )  ->  E. f  e.  ( II  Cn  K
) ( ( f `
 0 )  =  O  /\  ( f `
 1 )  =  O  /\  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  P ) )
9021, 24, 27, 77, 89syl13anc 1232 . . 3  |-  ( ph  ->  E. f  e.  ( II  Cn  K ) ( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  O  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  P ) )
911, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem4 29619 . . . 4  |-  ( (
ph  /\  O  e.  Y )  ->  (
( H `  O
)  =  P  <->  E. f  e.  ( II  Cn  K
) ( ( f `
 0 )  =  O  /\  ( f `
 1 )  =  O  /\  ( (
iota_ g  e.  (
II  Cn  C )
( ( F  o.  g )  =  ( G  o.  f )  /\  ( g ` 
0 )  =  P ) ) `  1
)  =  P ) ) )
926, 91mpdan 666 . . 3  |-  ( ph  ->  ( ( H `  O )  =  P  <->  E. f  e.  (
II  Cn  K )
( ( f ` 
0 )  =  O  /\  ( f ` 
1 )  =  O  /\  ( ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  f )  /\  (
g `  0 )  =  P ) ) ` 
1 )  =  P ) ) )
9390, 92mpbird 232 . 2  |-  ( ph  ->  ( H `  O
)  =  P )
94 coeq2 4982 . . . . 5  |-  ( f  =  H  ->  ( F  o.  f )  =  ( F  o.  H ) )
9594eqeq1d 2404 . . . 4  |-  ( f  =  H  ->  (
( F  o.  f
)  =  G  <->  ( F  o.  H )  =  G ) )
96 fveq1 5848 . . . . 5  |-  ( f  =  H  ->  (
f `  O )  =  ( H `  O ) )
9796eqeq1d 2404 . . . 4  |-  ( f  =  H  ->  (
( f `  O
)  =  P  <->  ( H `  O )  =  P ) )
9895, 97anbi12d 709 . . 3  |-  ( f  =  H  ->  (
( ( F  o.  f )  =  G  /\  ( f `  O )  =  P )  <->  ( ( F  o.  H )  =  G  /\  ( H `
 O )  =  P ) ) )
9998rspcev 3160 . 2  |-  ( ( H  e.  ( K  Cn  C )  /\  ( ( F  o.  H )  =  G  /\  ( H `  O )  =  P ) )  ->  E. f  e.  ( K  Cn  C
) ( ( F  o.  f )  =  G  /\  ( f `
 O )  =  P ) )
10012, 13, 93, 99syl12anc 1228 1  |-  ( ph  ->  E. f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   E!wreu 2756   {crab 2758    \ cdif 3411    i^i cin 3413   (/)c0 3738   ~Pcpw 3955   {csn 3972   U.cuni 4191    |-> cmpt 4453    X. cxp 4821   `'ccnv 4822    |` cres 4825   "cima 4826    o. ccom 4827    Fn wfn 5564   -->wf 5565   ` cfv 5569   iota_crio 6239  (class class class)co 6278   0cc0 9522   1c1 9523   [,]cicc 11585   ↾t crest 15035   Topctop 19686  TopOnctopon 19687    Cn ccn 20018  𝑛Locally cnlly 20258   Homeochmeo 20546   IIcii 21671  PConcpcon 29516  SConcscon 29517   CovMap ccvm 29552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-ec 7350  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-cn 20021  df-cnp 20022  df-cmp 20180  df-con 20205  df-lly 20259  df-nlly 20260  df-tx 20355  df-hmeo 20548  df-xms 21115  df-ms 21116  df-tms 21117  df-ii 21673  df-htpy 21762  df-phtpy 21763  df-phtpc 21784  df-pco 21797  df-pcon 29518  df-scon 29519  df-cvm 29553
This theorem is referenced by:  cvmlift3  29625
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