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Theorem cvmlift3lem9 24967
Description: Lemma for cvmlift2 24956. (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 24966 . 2  |-  ( ph  ->  H  e.  ( K  Cn  C ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem5 24963 . 2  |-  ( ph  ->  ( F  o.  H
)  =  G )
14 iitopon 18862 . . . . . 6  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
1514a1i 11 . . . . 5  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
16 scontop 24868 . . . . . . 7  |-  ( K  e. SCon  ->  K  e.  Top )
174, 16syl 16 . . . . . 6  |-  ( ph  ->  K  e.  Top )
182toptopon 16953 . . . . . 6  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
1917, 18sylib 189 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
20 cnconst2 17301 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  { O } )  e.  ( II  Cn  K ) )
22 0elunit 10971 . . . . 5  |-  0  e.  ( 0 [,] 1
)
23 fvconst2g 5904 . . . . 5  |-  ( ( O  e.  Y  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O )
246, 22, 23sylancl 644 . . . 4  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 0 )  =  O )
25 1elunit 10972 . . . . 5  |-  1  e.  ( 0 [,] 1
)
26 fvconst2g 5904 . . . . 5  |-  ( ( O  e.  Y  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { O }
) `  1 )  =  O )
276, 25, 26sylancl 644 . . . 4  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 1 )  =  O )
289sneqd 3787 . . . . . . . . 9  |-  ( ph  ->  { ( F `  P ) }  =  { ( G `  O ) } )
2928xpeq2d 4861 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( F `  P
) } )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
30 cvmcn 24902 . . . . . . . . . . . 12  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
313, 30syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C  Cn  J ) )
32 eqid 2404 . . . . . . . . . . . 12  |-  U. J  =  U. J
331, 32cnf 17264 . . . . . . . . . . 11  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
3431, 33syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : B --> U. J
)
35 ffn 5550 . . . . . . . . . 10  |-  ( F : B --> U. J  ->  F  Fn  B )
3634, 35syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  B )
37 fcoconst 5864 . . . . . . . . 9  |-  ( ( F  Fn  B  /\  P  e.  B )  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  P
) } ) )
3836, 8, 37syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  P
) } ) )
392, 32cnf 17264 . . . . . . . . . . 11  |-  ( G  e.  ( K  Cn  J )  ->  G : Y --> U. J )
407, 39syl 16 . . . . . . . . . 10  |-  ( ph  ->  G : Y --> U. J
)
41 ffn 5550 . . . . . . . . . 10  |-  ( G : Y --> U. J  ->  G  Fn  Y )
4240, 41syl 16 . . . . . . . . 9  |-  ( ph  ->  G  Fn  Y )
43 fcoconst 5864 . . . . . . . . 9  |-  ( ( G  Fn  Y  /\  O  e.  Y )  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
4442, 6, 43syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
4529, 38, 443eqtr4d 2446 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) ) )
46 fvconst2g 5904 . . . . . . . 8  |-  ( ( P  e.  B  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { P }
) `  0 )  =  P )
478, 22, 46sylancl 644 . . . . . . 7  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 0 )  =  P )
48 cvmtop1 24900 . . . . . . . . . . 11  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
493, 48syl 16 . . . . . . . . . 10  |-  ( ph  ->  C  e.  Top )
501toptopon 16953 . . . . . . . . . 10  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
5149, 50sylib 189 . . . . . . . . 9  |-  ( ph  ->  C  e.  (TopOn `  B ) )
52 cnconst2 17301 . . . . . . . . 9  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  C  e.  (TopOn `  B )  /\  P  e.  B
)  ->  ( (
0 [,] 1 )  X.  { P }
)  e.  ( II 
Cn  C ) )
5315, 51, 8, 52syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  { P } )  e.  ( II  Cn  C ) )
54 cvmtop2 24901 . . . . . . . . . . . . 13  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
553, 54syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  Top )
5632toptopon 16953 . . . . . . . . . . . 12  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
5755, 56sylib 189 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
5840, 6ffvelrnd 5830 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  O
)  e.  U. J
)
59 cnconst2 17301 . . . . . . . . . . 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 ) )
6015, 57, 58, 59syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( G `  O
) } )  e.  ( II  Cn  J
) )
6144, 60eqeltrd 2478 . . . . . . . . 9  |-  ( ph  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  e.  ( II  Cn  J
) )
62 fvconst2g 5904 . . . . . . . . . . 11  |-  ( ( ( G `  O
)  e.  U. J  /\  0  e.  (
0 [,] 1 ) )  ->  ( (
( 0 [,] 1
)  X.  { ( G `  O ) } ) `  0
)  =  ( G `
 O ) )
6358, 22, 62sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ ( G `  O ) } ) `
 0 )  =  ( G `  O
) )
6444fveq1d 5689 . . . . . . . . . 10  |-  ( ph  ->  ( ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) ) ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) ` 
0 ) )
6563, 64, 93eqtr4rd 2447 . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  =  ( ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) `  0
) )
661cvmlift 24939 . . . . . . . . 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 ) )
673, 61, 8, 65, 66syl22anc 1185 . . . . . . . 8  |-  ( ph  ->  E! g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )
68 coeq2 4990 . . . . . . . . . . 11  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( F  o.  g )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { P }
) ) )
6968eqeq1d 2412 . . . . . . . . . 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 } ) ) ) )
70 fveq1 5686 . . . . . . . . . . 11  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( g ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  { P } ) `  0
) )
7170eqeq1d 2412 . . . . . . . . . 10  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( ( g `
 0 )  =  P  <->  ( ( ( 0 [,] 1 )  X.  { P }
) `  0 )  =  P ) )
7269, 71anbi12d 692 . . . . . . . . 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 ) ) )
7372riota2 6531 . . . . . . . 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 }
) ) )
7453, 67, 73syl2anc 643 . . . . . . 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 } ) ) )
7545, 47, 74mpbi2and 888 . . . . . 6  |-  ( ph  ->  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )  =  ( ( 0 [,] 1 )  X.  { P } ) )
7675fveq1d 5689 . . . . 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 )
)
77 fvconst2g 5904 . . . . . 6  |-  ( ( P  e.  B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { P }
) `  1 )  =  P )
788, 25, 77sylancl 644 . . . . 5  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 1 )  =  P )
7976, 78eqtrd 2436 . . . 4  |-  ( ph  ->  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) `  1 )  =  P )
80 fveq1 5686 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( f ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  { O } ) `  0
) )
8180eqeq1d 2412 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( f `
 0 )  =  O  <->  ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O ) )
82 fveq1 5686 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( f ` 
1 )  =  ( ( ( 0 [,] 1 )  X.  { O } ) `  1
) )
8382eqeq1d 2412 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( f `
 1 )  =  O  <->  ( ( ( 0 [,] 1 )  X.  { O }
) `  1 )  =  O ) )
84 coeq2 4990 . . . . . . . . . . 11  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( G  o.  f )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) )
8584eqeq2d 2415 . . . . . . . . . 10  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( F  o.  g )  =  ( G  o.  f
)  <->  ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) ) )
8685anbi1d 686 . . . . . . . . 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 ) ) )
8786riotabidv 6510 . . . . . . . 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 ) ) )
8887fveq1d 5689 . . . . . . 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 ) )
8988eqeq1d 2412 . . . . . 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 ) )
9081, 83, 893anbi123d 1254 . . . . 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 ) ) )
9190rspcev 3012 . . . 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 ) )
9221, 24, 27, 79, 91syl13anc 1186 . . 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 ) )
931, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem4 24962 . . . 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 ) ) )
946, 93mpdan 650 . . 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 ) ) )
9592, 94mpbird 224 . 2  |-  ( ph  ->  ( H `  O
)  =  P )
96 coeq2 4990 . . . . 5  |-  ( f  =  H  ->  ( F  o.  f )  =  ( F  o.  H ) )
9796eqeq1d 2412 . . . 4  |-  ( f  =  H  ->  (
( F  o.  f
)  =  G  <->  ( F  o.  H )  =  G ) )
98 fveq1 5686 . . . . 5  |-  ( f  =  H  ->  (
f `  O )  =  ( H `  O ) )
9998eqeq1d 2412 . . . 4  |-  ( f  =  H  ->  (
( f `  O
)  =  P  <->  ( H `  O )  =  P ) )
10097, 99anbi12d 692 . . 3  |-  ( f  =  H  ->  (
( ( F  o.  f )  =  G  /\  ( f `  O )  =  P )  <->  ( ( F  o.  H )  =  G  /\  ( H `
 O )  =  P ) ) )
101100rspcev 3012 . 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 ) )
10212, 13, 95, 101syl12anc 1182 1  |-  ( ph  ->  E. f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   E!wreu 2668   {crab 2670    \ cdif 3277    i^i cin 3279   (/)c0 3588   ~Pcpw 3759   {csn 3774   U.cuni 3975    e. cmpt 4226    X. cxp 4835   `'ccnv 4836    |` cres 4839   "cima 4840    o. ccom 4841    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   iota_crio 6501   0cc0 8946   1c1 8947   [,]cicc 10875   ↾t crest 13603   Topctop 16913  TopOnctopon 16914    Cn ccn 17242  𝑛Locally cnlly 17481    Homeo chmeo 17738   IIcii 18858  PConcpcon 24859  SConcscon 24860   CovMap ccvm 24895
This theorem is referenced by:  cvmlift3  24968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-cn 17245  df-cnp 17246  df-cmp 17404  df-con 17428  df-lly 17482  df-nlly 17483  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-ii 18860  df-htpy 18948  df-phtpy 18949  df-phtpc 18970  df-pco 18983  df-pcon 24861  df-scon 24862  df-cvm 24896
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