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Theorem cvmlift3lem9 30043
Description: Lemma for cvmlift2 30032. (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 30042 . 2  |-  ( ph  ->  H  e.  ( K  Cn  C ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem5 30039 . 2  |-  ( ph  ->  ( F  o.  H
)  =  G )
14 iitopon 21904 . . . . . 6  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
1514a1i 11 . . . . 5  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
16 scontop 29944 . . . . . . 7  |-  ( K  e. SCon  ->  K  e.  Top )
174, 16syl 17 . . . . . 6  |-  ( ph  ->  K  e.  Top )
182toptopon 19941 . . . . . 6  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
1917, 18sylib 200 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
20 cnconst2 20292 . . . . 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 1267 . . . 4  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  { O } )  e.  ( II  Cn  K ) )
22 0elunit 11747 . . . . 5  |-  0  e.  ( 0 [,] 1
)
23 fvconst2g 6116 . . . . 5  |-  ( ( O  e.  Y  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O )
246, 22, 23sylancl 667 . . . 4  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 0 )  =  O )
25 1elunit 11748 . . . . 5  |-  1  e.  ( 0 [,] 1
)
26 fvconst2g 6116 . . . . 5  |-  ( ( O  e.  Y  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { O }
) `  1 )  =  O )
276, 25, 26sylancl 667 . . . 4  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 1 )  =  O )
289sneqd 3979 . . . . . . . . 9  |-  ( ph  ->  { ( F `  P ) }  =  { ( G `  O ) } )
2928xpeq2d 4857 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( F `  P
) } )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
30 cvmcn 29978 . . . . . . . . . 10  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
31 eqid 2450 . . . . . . . . . . 11  |-  U. J  =  U. J
321, 31cnf 20255 . . . . . . . . . 10  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
33 ffn 5726 . . . . . . . . . 10  |-  ( F : B --> U. J  ->  F  Fn  B )
343, 30, 32, 334syl 19 . . . . . . . . 9  |-  ( ph  ->  F  Fn  B )
35 fcoconst 6058 . . . . . . . . 9  |-  ( ( F  Fn  B  /\  P  e.  B )  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  P
) } ) )
3634, 8, 35syl2anc 666 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  P
) } ) )
372, 31cnf 20255 . . . . . . . . . . 11  |-  ( G  e.  ( K  Cn  J )  ->  G : Y --> U. J )
387, 37syl 17 . . . . . . . . . 10  |-  ( ph  ->  G : Y --> U. J
)
39 ffn 5726 . . . . . . . . . 10  |-  ( G : Y --> U. J  ->  G  Fn  Y )
4038, 39syl 17 . . . . . . . . 9  |-  ( ph  ->  G  Fn  Y )
41 fcoconst 6058 . . . . . . . . 9  |-  ( ( G  Fn  Y  /\  O  e.  Y )  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
4240, 6, 41syl2anc 666 . . . . . . . 8  |-  ( ph  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
4329, 36, 423eqtr4d 2494 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) ) )
44 fvconst2g 6116 . . . . . . . 8  |-  ( ( P  e.  B  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { P }
) `  0 )  =  P )
458, 22, 44sylancl 667 . . . . . . 7  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 0 )  =  P )
46 cvmtop1 29976 . . . . . . . . . . 11  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
473, 46syl 17 . . . . . . . . . 10  |-  ( ph  ->  C  e.  Top )
481toptopon 19941 . . . . . . . . . 10  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
4947, 48sylib 200 . . . . . . . . 9  |-  ( ph  ->  C  e.  (TopOn `  B ) )
50 cnconst2 20292 . . . . . . . . 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 1267 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  { P } )  e.  ( II  Cn  C ) )
52 cvmtop2 29977 . . . . . . . . . . . . 13  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
533, 52syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  Top )
5431toptopon 19941 . . . . . . . . . . . 12  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
5553, 54sylib 200 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
5638, 6ffvelrnd 6021 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  O
)  e.  U. J
)
57 cnconst2 20292 . . . . . . . . . . 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 1267 . . . . . . . . . 10  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( G `  O
) } )  e.  ( II  Cn  J
) )
5942, 58eqeltrd 2528 . . . . . . . . 9  |-  ( ph  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  e.  ( II  Cn  J
) )
60 fvconst2g 6116 . . . . . . . . . . 11  |-  ( ( ( G `  O
)  e.  U. J  /\  0  e.  (
0 [,] 1 ) )  ->  ( (
( 0 [,] 1
)  X.  { ( G `  O ) } ) `  0
)  =  ( G `
 O ) )
6156, 22, 60sylancl 667 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ ( G `  O ) } ) `
 0 )  =  ( G `  O
) )
6242fveq1d 5865 . . . . . . . . . 10  |-  ( ph  ->  ( ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) ) ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) ` 
0 ) )
6361, 62, 93eqtr4rd 2495 . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  =  ( ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) `  0
) )
641cvmlift 30015 . . . . . . . . 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 1268 . . . . . . . 8  |-  ( ph  ->  E! g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )
66 coeq2 4992 . . . . . . . . . . 11  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( F  o.  g )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { P }
) ) )
6766eqeq1d 2452 . . . . . . . . . 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 5862 . . . . . . . . . . 11  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( g ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  { P } ) `  0
) )
6968eqeq1d 2452 . . . . . . . . . 10  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( ( g `
 0 )  =  P  <->  ( ( ( 0 [,] 1 )  X.  { P }
) `  0 )  =  P ) )
7067, 69anbi12d 716 . . . . . . . . 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 6272 . . . . . . . 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 666 . . . . . . 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 931 . . . . . 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 5865 . . . . 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 6116 . . . . . 6  |-  ( ( P  e.  B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { P }
) `  1 )  =  P )
768, 25, 75sylancl 667 . . . . 5  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 1 )  =  P )
7774, 76eqtrd 2484 . . . 4  |-  ( ph  ->  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) `  1 )  =  P )
78 fveq1 5862 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( f ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  { O } ) `  0
) )
7978eqeq1d 2452 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( f `
 0 )  =  O  <->  ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O ) )
80 fveq1 5862 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( f ` 
1 )  =  ( ( ( 0 [,] 1 )  X.  { O } ) `  1
) )
8180eqeq1d 2452 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( f `
 1 )  =  O  <->  ( ( ( 0 [,] 1 )  X.  { O }
) `  1 )  =  O ) )
82 coeq2 4992 . . . . . . . . . . 11  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( G  o.  f )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) )
8382eqeq2d 2460 . . . . . . . . . 10  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( F  o.  g )  =  ( G  o.  f
)  <->  ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) ) )
8483anbi1d 710 . . . . . . . . 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 6252 . . . . . . . 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 5865 . . . . . . 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 2452 . . . . . 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 1338 . . . . 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 3149 . . . 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 1269 . . 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 30038 . . . 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 673 . . 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 236 . 2  |-  ( ph  ->  ( H `  O
)  =  P )
94 coeq2 4992 . . . . 5  |-  ( f  =  H  ->  ( F  o.  f )  =  ( F  o.  H ) )
9594eqeq1d 2452 . . . 4  |-  ( f  =  H  ->  (
( F  o.  f
)  =  G  <->  ( F  o.  H )  =  G ) )
96 fveq1 5862 . . . . 5  |-  ( f  =  H  ->  (
f `  O )  =  ( H `  O ) )
9796eqeq1d 2452 . . . 4  |-  ( f  =  H  ->  (
( f `  O
)  =  P  <->  ( H `  O )  =  P ) )
9895, 97anbi12d 716 . . 3  |-  ( f  =  H  ->  (
( ( F  o.  f )  =  G  /\  ( f `  O )  =  P )  <->  ( ( F  o.  H )  =  G  /\  ( H `
 O )  =  P ) ) )
9998rspcev 3149 . 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 1265 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 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736   E.wrex 2737   E!wreu 2738   {crab 2740    \ cdif 3400    i^i cin 3402   (/)c0 3730   ~Pcpw 3950   {csn 3967   U.cuni 4197    |-> cmpt 4460    X. cxp 4831   `'ccnv 4832    |` cres 4835   "cima 4836    o. ccom 4837    Fn wfn 5576   -->wf 5577   ` cfv 5581   iota_crio 6249  (class class class)co 6288   0cc0 9536   1c1 9537   [,]cicc 11635   ↾t crest 15312   Topctop 19910  TopOnctopon 19911    Cn ccn 20233  𝑛Locally cnlly 20473   Homeochmeo 20761   IIcii 21900  PConcpcon 29935  SConcscon 29936   CovMap ccvm 29971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-ec 7362  df-map 7471  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-cn 20236  df-cnp 20237  df-cmp 20395  df-con 20420  df-lly 20474  df-nlly 20475  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330  df-ii 21902  df-htpy 21994  df-phtpy 21995  df-phtpc 22016  df-pco 22029  df-pcon 29937  df-scon 29938  df-cvm 29972
This theorem is referenced by:  cvmlift3  30044
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