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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmlift3lem9 Structured version   Visualization version   Unicode version
Theorem cvmlift3lem9 31309
Description: Lemma for cvmlift2 31298. (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. SConn )
cvmlift3.l |- ( ph -> K e. 𝑛Locally PConn )
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. ( ( C ↾t c ) Homeo ( J ↾t 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. SConn )
5 cvmlift3.l . . 3 |- ( ph -> K e. 𝑛Locally PConn )
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. ( ( C ↾t c ) Homeo ( J ↾t k ) ) ) ) } )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cvmlift3lem8 31308 . 2 |- ( ph -> H e. ( K Cn C ) )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10cvmlift3lem5 31305 . 2 |- ( ph -> ( F o. H ) = G )
14 iitopon 22682 . . . . . 6 |- II e. (TopOn ` ( 0 [,] 1 ) )
1514a1i 11 . . . . 5 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
16 sconntop 31210 . . . . . . 7 |- ( K e. SConn -> K e. Top )
174, 16syl 17 . . . . . 6 |- ( ph -> K e. Top )
182toptopon 20722 . . . . . 6 |- ( K e. Top <-> K e. (TopOn ` Y ) )
1917, 18sylib 208 . . . . 5 |- ( ph -> K e. (TopOn ` Y ) )
20 cnconst2 21087 . . . . 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 1326 . . . 4 |- ( ph -> ( ( 0 [,] 1 ) X. { O } ) e. ( II Cn K ) )
22 0elunit 12290 . . . . 5 |- 0 e. ( 0 [,] 1 )
23 fvconst2g 6467 . . . . 5 |- ( ( O e. Y /\ 0 e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { O } ) ` 0 ) = O )
246, 22, 23sylancl 694 . . . 4 |- ( ph -> ( ( ( 0 [,] 1 ) X. { O } ) ` 0 ) = O )
25 1elunit 12291 . . . . 5 |- 1 e. ( 0 [,] 1 )
26 fvconst2g 6467 . . . . 5 |- ( ( O e. Y /\ 1 e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { O } ) ` 1 ) = O )
276, 25, 26sylancl 694 . . . 4 |- ( ph -> ( ( ( 0 [,] 1 ) X. { O } ) ` 1 ) = O )
289sneqd 4189 . . . . . . . . 9 |- ( ph -> { ( F ` P ) } = { ( G ` O ) } )
2928xpeq2d 5139 . . . . . . . 8 |- ( ph -> ( ( 0 [,] 1 ) X. { ( F ` P ) } ) = ( ( 0 [,] 1 ) X. { ( G ` O ) } ) )
30 cvmcn 31244 . . . . . . . . . 10 |- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) )
31 eqid 2622 . . . . . . . . . . 11 |- U. J = U. J
321, 31cnf 21050 . . . . . . . . . 10 |- ( F e. ( C Cn J ) -> F : B --> U. J )
33 ffn 6045 . . . . . . . . . 10 |- ( F : B --> U. J -> F Fn B )
343, 30, 32, 334syl 19 . . . . . . . . 9 |- ( ph -> F Fn B )
35 fcoconst 6401 . . . . . . . . 9 |- ( ( F Fn B /\ P e. B ) -> ( F o. ( ( 0 [,] 1 ) X. { P } ) ) = ( ( 0 [,] 1 ) X. { ( F ` P ) } ) )
3634, 8, 35syl2anc 693 . . . . . . . 8 |- ( ph -> ( F o. ( ( 0 [,] 1 ) X. { P } ) ) = ( ( 0 [,] 1 ) X. { ( F ` P ) } ) )
372, 31cnf 21050 . . . . . . . . . . 11 |- ( G e. ( K Cn J ) -> G : Y --> U. J )
387, 37syl 17 . . . . . . . . . 10 |- ( ph -> G : Y --> U. J )
39 ffn 6045 . . . . . . . . . 10 |- ( G : Y --> U. J -> G Fn Y )
4038, 39syl 17 . . . . . . . . 9 |- ( ph -> G Fn Y )
41 fcoconst 6401 . . . . . . . . 9 |- ( ( G Fn Y /\ O e. Y ) -> ( G o. ( ( 0 [,] 1 ) X. { O } ) ) = ( ( 0 [,] 1 ) X. { ( G ` O ) } ) )
4240, 6, 41syl2anc 693 . . . . . . . 8 |- ( ph -> ( G o. ( ( 0 [,] 1 ) X. { O } ) ) = ( ( 0 [,] 1 ) X. { ( G ` O ) } ) )
4329, 36, 423eqtr4d 2666 . . . . . . 7 |- ( ph -> ( F o. ( ( 0 [,] 1 ) X. { P } ) ) = ( G o. ( ( 0 [,] 1 ) X. { O } ) ) )
44 fvconst2g 6467 . . . . . . . 8 |- ( ( P e. B /\ 0 e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { P } ) ` 0 ) = P )
458, 22, 44sylancl 694 . . . . . . 7 |- ( ph -> ( ( ( 0 [,] 1 ) X. { P } ) ` 0 ) = P )
46 cvmtop1 31242 . . . . . . . . . . 11 |- ( F e. ( C CovMap J ) -> C e. Top )
473, 46syl 17 . . . . . . . . . 10 |- ( ph -> C e. Top )
481toptopon 20722 . . . . . . . . . 10 |- ( C e. Top <-> C e. (TopOn ` B ) )
4947, 48sylib 208 . . . . . . . . 9 |- ( ph -> C e. (TopOn ` B ) )
50 cnconst2 21087 . . . . . . . . 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 1326 . . . . . . . 8 |- ( ph -> ( ( 0 [,] 1 ) X. { P } ) e. ( II Cn C ) )
52 cvmtop2 31243 . . . . . . . . . . . . 13 |- ( F e. ( C CovMap J ) -> J e. Top )
533, 52syl 17 . . . . . . . . . . . 12 |- ( ph -> J e. Top )
5431toptopon 20722 . . . . . . . . . . . 12 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
5553, 54sylib 208 . . . . . . . . . . 11 |- ( ph -> J e. (TopOn ` U. J ) )
5638, 6ffvelrnd 6360 . . . . . . . . . . 11 |- ( ph -> ( G ` O ) e. U. J )
57 cnconst2 21087 . . . . . . . . . . 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 1326 . . . . . . . . . 10 |- ( ph -> ( ( 0 [,] 1 ) X. { ( G ` O ) } ) e. ( II Cn J ) )
5942, 58eqeltrd 2701 . . . . . . . . 9 |- ( ph -> ( G o. ( ( 0 [,] 1 ) X. { O } ) ) e. ( II Cn J ) )
60 fvconst2g 6467 . . . . . . . . . . 11 |- ( ( ( G ` O ) e. U. J /\ 0 e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { ( G ` O ) } ) ` 0 ) = ( G ` O ) )
6156, 22, 60sylancl 694 . . . . . . . . . 10 |- ( ph -> ( ( ( 0 [,] 1 ) X. { ( G ` O ) } ) ` 0 ) = ( G ` O ) )
6242fveq1d 6193 . . . . . . . . . 10 |- ( ph -> ( ( G o. ( ( 0 [,] 1 ) X. { O } ) ) ` 0 ) = ( ( ( 0 [,] 1 ) X. { ( G ` O ) } ) ` 0 ) )
6361, 62, 93eqtr4rd 2667 . . . . . . . . 9 |- ( ph -> ( F ` P ) = ( ( G o. ( ( 0 [,] 1 ) X. { O } ) ) ` 0 ) )
641cvmlift 31281 . . . . . . . . 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 1327 . . . . . . . 8 |- ( ph -> E! g e. ( II Cn C ) ( ( F o. g ) = ( G o. ( ( 0 [,] 1 ) X. { O } ) ) /\ ( g ` 0 ) = P ) )
66 coeq2 5280 . . . . . . . . . . 11 |- ( g = ( ( 0 [,] 1 ) X. { P } ) -> ( F o. g ) = ( F o. ( ( 0 [,] 1 ) X. { P } ) ) )
6766eqeq1d 2624 . . . . . . . . . 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 6190 . . . . . . . . . . 11 |- ( g = ( ( 0 [,] 1 ) X. { P } ) -> ( g ` 0 ) = ( ( ( 0 [,] 1 ) X. { P } ) ` 0 ) )
6968eqeq1d 2624 . . . . . . . . . 10 |- ( g = ( ( 0 [,] 1 ) X. { P } ) -> ( ( g ` 0 ) = P <-> ( ( ( 0 [,] 1 ) X. { P } ) ` 0 ) = P ) )
7067, 69anbi12d 747 . . . . . . . . 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 6633 . . . . . . . 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 693 . . . . . . 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 956 . . . . . 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 6193 . . . . 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 6467 . . . . . 6 |- ( ( P e. B /\ 1 e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { P } ) ` 1 ) = P )
768, 25, 75sylancl 694 . . . . 5 |- ( ph -> ( ( ( 0 [,] 1 ) X. { P } ) ` 1 ) = P )
7774, 76eqtrd 2656 . . . 4 |- ( ph -> ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. ( ( 0 [,] 1 ) X. { O } ) ) /\ ( g ` 0 ) = P ) ) ` 1 ) = P )
78 fveq1 6190 . . . . . . 7 |- ( f = ( ( 0 [,] 1 ) X. { O } ) -> ( f ` 0 ) = ( ( ( 0 [,] 1 ) X. { O } ) ` 0 ) )
7978eqeq1d 2624 . . . . . 6 |- ( f = ( ( 0 [,] 1 ) X. { O } ) -> ( ( f ` 0 ) = O <-> ( ( ( 0 [,] 1 ) X. { O } ) ` 0 ) = O ) )
80 fveq1 6190 . . . . . . 7 |- ( f = ( ( 0 [,] 1 ) X. { O } ) -> ( f ` 1 ) = ( ( ( 0 [,] 1 ) X. { O } ) ` 1 ) )
8180eqeq1d 2624 . . . . . 6 |- ( f = ( ( 0 [,] 1 ) X. { O } ) -> ( ( f ` 1 ) = O <-> ( ( ( 0 [,] 1 ) X. { O } ) ` 1 ) = O ) )
82 coeq2 5280 . . . . . . . . . . 11 |- ( f = ( ( 0 [,] 1 ) X. { O } ) -> ( G o. f ) = ( G o. ( ( 0 [,] 1 ) X. { O } ) ) )
8382eqeq2d 2632 . . . . . . . . . 10 |- ( f = ( ( 0 [,] 1 ) X. { O } ) -> ( ( F o. g ) = ( G o. f ) <-> ( F o. g ) = ( G o. ( ( 0 [,] 1 ) X. { O } ) ) ) )
8483anbi1d 741 . . . . . . . . 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 6613 . . . . . . . 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 6193 . . . . . . 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 2624 . . . . . 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 1399 . . . . 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 3309 . . . 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 1328 . . 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 31304 . . . 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 702 . . 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 247 . 2 |- ( ph -> ( H ` O ) = P )
94 coeq2 5280 . . . . 5 |- ( f = H -> ( F o. f ) = ( F o. H ) )
9594eqeq1d 2624 . . . 4 |- ( f = H -> ( ( F o. f ) = G <-> ( F o. H ) = G ) )
96 fveq1 6190 . . . . 5 |- ( f = H -> ( f ` O ) = ( H ` O ) )
9796eqeq1d 2624 . . . 4 |- ( f = H -> ( ( f ` O ) = P <-> ( H ` O ) = P ) )
9895, 97anbi12d 747 . . 3 |- ( f = H -> ( ( ( F o. f ) = G /\ ( f ` O ) = P ) <-> ( ( F o. H ) = G /\ ( H ` O ) = P ) ) )
9998rspcev 3309 . 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 1324 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   {crab 2916   \ cdif 3571   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888   iota_crio 6610  (class class class)co 6650   0cc0 9936   1c1 9937   [,]cicc 12178   ↾t crest 16081   Topctop 20698  TopOnctopon 20715   Cn ccn 21028  𝑛Locally cnlly 21268   Homeochmeo 21556   IIcii 22678  PConncpconn 31201  SConncsconn 31202   CovMap ccvm 31237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-cnp 21032  df-cmp 21190  df-conn 21215  df-lly 21269  df-nlly 21270  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791  df-pco 22805  df-pconn 31203  df-sconn 31204  df-cvm 31238
This theorem is referenced by:  cvmlift3  31310
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