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Theorem cvmlift3lem9 23858
Description: Lemma for cvmlift2 23847. (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 23857 . 2  |-  ( ph  ->  H  e.  ( K  Cn  C ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem5 23854 . 2  |-  ( ph  ->  ( F  o.  H
)  =  G )
14 iitopon 18383 . . . . . 6  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
1514a1i 10 . . . . 5  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
16 scontop 23759 . . . . . . 7  |-  ( K  e. SCon  ->  K  e.  Top )
174, 16syl 15 . . . . . 6  |-  ( ph  ->  K  e.  Top )
182toptopon 16671 . . . . . 6  |-  ( K  e.  Top  <->  K  e.  (TopOn `  Y ) )
1917, 18sylib 188 . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
20 cnconst2 17011 . . . . 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 1182 . . . 4  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  { O } )  e.  ( II  Cn  K ) )
22 0elunit 10754 . . . . 5  |-  0  e.  ( 0 [,] 1
)
23 fvconst2g 5727 . . . . 5  |-  ( ( O  e.  Y  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O )
246, 22, 23sylancl 643 . . . 4  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 0 )  =  O )
25 1elunit 10755 . . . . 5  |-  1  e.  ( 0 [,] 1
)
26 fvconst2g 5727 . . . . 5  |-  ( ( O  e.  Y  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { O }
) `  1 )  =  O )
276, 25, 26sylancl 643 . . . 4  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ O } ) `
 1 )  =  O )
289sneqd 3653 . . . . . . . . 9  |-  ( ph  ->  { ( F `  P ) }  =  { ( G `  O ) } )
2928xpeq2d 4713 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( F `  P
) } )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
30 cvmcn 23793 . . . . . . . . . . . 12  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
313, 30syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C  Cn  J ) )
32 eqid 2283 . . . . . . . . . . . 12  |-  U. J  =  U. J
331, 32cnf 16976 . . . . . . . . . . 11  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
3431, 33syl 15 . . . . . . . . . 10  |-  ( ph  ->  F : B --> U. J
)
35 ffn 5389 . . . . . . . . . 10  |-  ( F : B --> U. J  ->  F  Fn  B )
3634, 35syl 15 . . . . . . . . 9  |-  ( ph  ->  F  Fn  B )
37 fcoconst 5695 . . . . . . . . 9  |-  ( ( F  Fn  B  /\  P  e.  B )  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  P
) } ) )
3836, 8, 37syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( ( 0 [,] 1 )  X.  {
( F `  P
) } ) )
392, 32cnf 16976 . . . . . . . . . . 11  |-  ( G  e.  ( K  Cn  J )  ->  G : Y --> U. J )
407, 39syl 15 . . . . . . . . . 10  |-  ( ph  ->  G : Y --> U. J
)
41 ffn 5389 . . . . . . . . . 10  |-  ( G : Y --> U. J  ->  G  Fn  Y )
4240, 41syl 15 . . . . . . . . 9  |-  ( ph  ->  G  Fn  Y )
43 fcoconst 5695 . . . . . . . . 9  |-  ( ( G  Fn  Y  /\  O  e.  Y )  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
4442, 6, 43syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  =  ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) )
4529, 38, 443eqtr4d 2325 . . . . . . 7  |-  ( ph  ->  ( F  o.  (
( 0 [,] 1
)  X.  { P } ) )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) ) )
46 fvconst2g 5727 . . . . . . . 8  |-  ( ( P  e.  B  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { P }
) `  0 )  =  P )
478, 22, 46sylancl 643 . . . . . . 7  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 0 )  =  P )
48 cvmtop1 23791 . . . . . . . . . . 11  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
493, 48syl 15 . . . . . . . . . 10  |-  ( ph  ->  C  e.  Top )
501toptopon 16671 . . . . . . . . . 10  |-  ( C  e.  Top  <->  C  e.  (TopOn `  B ) )
5149, 50sylib 188 . . . . . . . . 9  |-  ( ph  ->  C  e.  (TopOn `  B ) )
52 cnconst2 17011 . . . . . . . . 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 1182 . . . . . . . 8  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  { P } )  e.  ( II  Cn  C ) )
54 cvmtop2 23792 . . . . . . . . . . . . 13  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
553, 54syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  Top )
5632toptopon 16671 . . . . . . . . . . . 12  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
5755, 56sylib 188 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
58 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( G : Y --> U. J  /\  O  e.  Y
)  ->  ( G `  O )  e.  U. J )
5940, 6, 58syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( G `  O
)  e.  U. J
)
60 cnconst2 17011 . . . . . . . . . . 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 ) )
6115, 57, 59, 60syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( G `  O
) } )  e.  ( II  Cn  J
) )
6244, 61eqeltrd 2357 . . . . . . . . 9  |-  ( ph  ->  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  e.  ( II  Cn  J
) )
63 fvconst2g 5727 . . . . . . . . . . 11  |-  ( ( ( G `  O
)  e.  U. J  /\  0  e.  (
0 [,] 1 ) )  ->  ( (
( 0 [,] 1
)  X.  { ( G `  O ) } ) `  0
)  =  ( G `
 O ) )
6459, 22, 63sylancl 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ ( G `  O ) } ) `
 0 )  =  ( G `  O
) )
6544fveq1d 5527 . . . . . . . . . 10  |-  ( ph  ->  ( ( G  o.  ( ( 0 [,] 1 )  X.  { O } ) ) ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  {
( G `  O
) } ) ` 
0 ) )
6664, 65, 93eqtr4rd 2326 . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  =  ( ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) `  0
) )
671cvmlift 23830 . . . . . . . . 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 ) )
683, 62, 8, 66, 67syl22anc 1183 . . . . . . . 8  |-  ( ph  ->  E! g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )
69 coeq2 4842 . . . . . . . . . . 11  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( F  o.  g )  =  ( F  o.  ( ( 0 [,] 1 )  X.  { P }
) ) )
7069eqeq1d 2291 . . . . . . . . . 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 } ) ) ) )
71 fveq1 5524 . . . . . . . . . . 11  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( g ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  { P } ) `  0
) )
7271eqeq1d 2291 . . . . . . . . . 10  |-  ( g  =  ( ( 0 [,] 1 )  X. 
{ P } )  ->  ( ( g `
 0 )  =  P  <->  ( ( ( 0 [,] 1 )  X.  { P }
) `  0 )  =  P ) )
7370, 72anbi12d 691 . . . . . . . . 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 ) ) )
7473riota2 6327 . . . . . . . 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 }
) ) )
7553, 68, 74syl2anc 642 . . . . . . 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 } ) ) )
7645, 47, 75mpbi2and 887 . . . . . 6  |-  ( ph  ->  ( iota_ g  e.  ( II  Cn  C ) ( ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) )  /\  (
g `  0 )  =  P ) )  =  ( ( 0 [,] 1 )  X.  { P } ) )
7776fveq1d 5527 . . . . 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 )
)
78 fvconst2g 5727 . . . . . 6  |-  ( ( P  e.  B  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( ( 0 [,] 1 )  X.  { P }
) `  1 )  =  P )
798, 25, 78sylancl 643 . . . . 5  |-  ( ph  ->  ( ( ( 0 [,] 1 )  X. 
{ P } ) `
 1 )  =  P )
8077, 79eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( iota_ g  e.  ( II  Cn  C
) ( ( F  o.  g )  =  ( G  o.  (
( 0 [,] 1
)  X.  { O } ) )  /\  ( g `  0
)  =  P ) ) `  1 )  =  P )
81 fveq1 5524 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( f ` 
0 )  =  ( ( ( 0 [,] 1 )  X.  { O } ) `  0
) )
8281eqeq1d 2291 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( f `
 0 )  =  O  <->  ( ( ( 0 [,] 1 )  X.  { O }
) `  0 )  =  O ) )
83 fveq1 5524 . . . . . . 7  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( f ` 
1 )  =  ( ( ( 0 [,] 1 )  X.  { O } ) `  1
) )
8483eqeq1d 2291 . . . . . 6  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( f `
 1 )  =  O  <->  ( ( ( 0 [,] 1 )  X.  { O }
) `  1 )  =  O ) )
85 coeq2 4842 . . . . . . . . . . 11  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( G  o.  f )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) )
8685eqeq2d 2294 . . . . . . . . . 10  |-  ( f  =  ( ( 0 [,] 1 )  X. 
{ O } )  ->  ( ( F  o.  g )  =  ( G  o.  f
)  <->  ( F  o.  g )  =  ( G  o.  ( ( 0 [,] 1 )  X.  { O }
) ) ) )
8786anbi1d 685 . . . . . . . . 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 ) ) )
8887riotabidv 6306 . . . . . . . 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 ) ) )
8988fveq1d 5527 . . . . . . 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 ) )
9089eqeq1d 2291 . . . . . 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 ) )
9182, 84, 903anbi123d 1252 . . . . 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 ) ) )
9291rspcev 2884 . . . 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 ) )
9321, 24, 27, 80, 92syl13anc 1184 . . 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 ) )
941, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem4 23853 . . . 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 ) ) )
956, 94mpdan 649 . . 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 ) ) )
9693, 95mpbird 223 . 2  |-  ( ph  ->  ( H `  O
)  =  P )
97 coeq2 4842 . . . . 5  |-  ( f  =  H  ->  ( F  o.  f )  =  ( F  o.  H ) )
9897eqeq1d 2291 . . . 4  |-  ( f  =  H  ->  (
( F  o.  f
)  =  G  <->  ( F  o.  H )  =  G ) )
99 fveq1 5524 . . . . 5  |-  ( f  =  H  ->  (
f `  O )  =  ( H `  O ) )
10099eqeq1d 2291 . . . 4  |-  ( f  =  H  ->  (
( f `  O
)  =  P  <->  ( H `  O )  =  P ) )
10198, 100anbi12d 691 . . 3  |-  ( f  =  H  ->  (
( ( F  o.  f )  =  G  /\  ( f `  O )  =  P )  <->  ( ( F  o.  H )  =  G  /\  ( H `
 O )  =  P ) ) )
102101rspcev 2884 . 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 ) )
10312, 13, 96, 102syl12anc 1180 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   E!wreu 2545   {crab 2547    \ cdif 3149    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827    e. cmpt 4077    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   iota_crio 6297   0cc0 8737   1c1 8738   [,]cicc 10659   ↾t crest 13325   Topctop 16631  TopOnctopon 16632    Cn ccn 16954  𝑛Locally cnlly 17191    Homeo chmeo 17444   IIcii 18379  PConcpcon 23750  SConcscon 23751   CovMap ccvm 23786
This theorem is referenced by:  cvmlift3  23859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cn 16957  df-cnp 16958  df-cmp 17114  df-con 17138  df-lly 17192  df-nlly 17193  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-ii 18381  df-htpy 18468  df-phtpy 18469  df-phtpc 18490  df-pco 18503  df-pcon 23752  df-scon 23753  df-cvm 23787
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