Step |
Hyp |
Ref |
Expression |
1 |
|
cvmlift2.b |
|- B = U. C |
2 |
|
cvmlift2.f |
|- ( ph -> F e. ( C CovMap J ) ) |
3 |
|
cvmlift2.g |
|- ( ph -> G e. ( ( II tX II ) Cn J ) ) |
4 |
|
cvmlift2.p |
|- ( ph -> P e. B ) |
5 |
|
cvmlift2.i |
|- ( ph -> ( F ` P ) = ( 0 G 0 ) ) |
6 |
|
cvmlift2.h |
|- H = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( z G 0 ) ) /\ ( f ` 0 ) = P ) ) |
7 |
|
cvmlift2.k |
|- K = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) ) |
8 |
|
cvmlift2lem10.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 ) ) ) ) } ) |
9 |
|
cvmlift2lem9.1 |
|- ( ph -> ( X G Y ) e. M ) |
10 |
|
cvmlift2lem9.2 |
|- ( ph -> T e. ( S ` M ) ) |
11 |
|
cvmlift2lem9.3 |
|- ( ph -> U e. II ) |
12 |
|
cvmlift2lem9.4 |
|- ( ph -> V e. II ) |
13 |
|
cvmlift2lem9.5 |
|- ( ph -> ( II |`t U ) e. Conn ) |
14 |
|
cvmlift2lem9.6 |
|- ( ph -> ( II |`t V ) e. Conn ) |
15 |
|
cvmlift2lem9.7 |
|- ( ph -> X e. U ) |
16 |
|
cvmlift2lem9.8 |
|- ( ph -> Y e. V ) |
17 |
|
cvmlift2lem9.9 |
|- ( ph -> ( U X. V ) C_ ( `' G " M ) ) |
18 |
|
cvmlift2lem9.10 |
|- ( ph -> Z e. V ) |
19 |
|
cvmlift2lem9.11 |
|- ( ph -> ( K |` ( U X. { Z } ) ) e. ( ( ( II tX II ) |`t ( U X. { Z } ) ) Cn C ) ) |
20 |
|
cvmlift2lem9.w |
|- W = ( iota_ b e. T ( X K Y ) e. b ) |
21 |
|
iitop |
|- II e. Top |
22 |
|
iiuni |
|- ( 0 [,] 1 ) = U. II |
23 |
21 21 22 22
|
txunii |
|- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II ) |
24 |
1 2 3 4 5 6 7
|
cvmlift2lem5 |
|- ( ph -> K : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B ) |
25 |
1 2 3 4 5 6 7
|
cvmlift2lem7 |
|- ( ph -> ( F o. K ) = G ) |
26 |
25 3
|
eqeltrd |
|- ( ph -> ( F o. K ) e. ( ( II tX II ) Cn J ) ) |
27 |
21 21
|
txtopi |
|- ( II tX II ) e. Top |
28 |
27
|
a1i |
|- ( ph -> ( II tX II ) e. Top ) |
29 |
|
elssuni |
|- ( U e. II -> U C_ U. II ) |
30 |
29 22
|
sseqtrrdi |
|- ( U e. II -> U C_ ( 0 [,] 1 ) ) |
31 |
11 30
|
syl |
|- ( ph -> U C_ ( 0 [,] 1 ) ) |
32 |
31 15
|
sseldd |
|- ( ph -> X e. ( 0 [,] 1 ) ) |
33 |
|
elssuni |
|- ( V e. II -> V C_ U. II ) |
34 |
33 22
|
sseqtrrdi |
|- ( V e. II -> V C_ ( 0 [,] 1 ) ) |
35 |
12 34
|
syl |
|- ( ph -> V C_ ( 0 [,] 1 ) ) |
36 |
35 16
|
sseldd |
|- ( ph -> Y e. ( 0 [,] 1 ) ) |
37 |
|
opelxpi |
|- ( ( X e. ( 0 [,] 1 ) /\ Y e. ( 0 [,] 1 ) ) -> <. X , Y >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
38 |
32 36 37
|
syl2anc |
|- ( ph -> <. X , Y >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
39 |
24 32 36
|
fovrnd |
|- ( ph -> ( X K Y ) e. B ) |
40 |
|
fvco3 |
|- ( ( K : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ <. X , Y >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( ( F o. K ) ` <. X , Y >. ) = ( F ` ( K ` <. X , Y >. ) ) ) |
41 |
24 38 40
|
syl2anc |
|- ( ph -> ( ( F o. K ) ` <. X , Y >. ) = ( F ` ( K ` <. X , Y >. ) ) ) |
42 |
25
|
fveq1d |
|- ( ph -> ( ( F o. K ) ` <. X , Y >. ) = ( G ` <. X , Y >. ) ) |
43 |
41 42
|
eqtr3d |
|- ( ph -> ( F ` ( K ` <. X , Y >. ) ) = ( G ` <. X , Y >. ) ) |
44 |
|
df-ov |
|- ( X K Y ) = ( K ` <. X , Y >. ) |
45 |
44
|
fveq2i |
|- ( F ` ( X K Y ) ) = ( F ` ( K ` <. X , Y >. ) ) |
46 |
|
df-ov |
|- ( X G Y ) = ( G ` <. X , Y >. ) |
47 |
43 45 46
|
3eqtr4g |
|- ( ph -> ( F ` ( X K Y ) ) = ( X G Y ) ) |
48 |
47 9
|
eqeltrd |
|- ( ph -> ( F ` ( X K Y ) ) e. M ) |
49 |
8 1 20
|
cvmsiota |
|- ( ( F e. ( C CovMap J ) /\ ( T e. ( S ` M ) /\ ( X K Y ) e. B /\ ( F ` ( X K Y ) ) e. M ) ) -> ( W e. T /\ ( X K Y ) e. W ) ) |
50 |
2 10 39 48 49
|
syl13anc |
|- ( ph -> ( W e. T /\ ( X K Y ) e. W ) ) |
51 |
44
|
eleq1i |
|- ( ( X K Y ) e. W <-> ( K ` <. X , Y >. ) e. W ) |
52 |
51
|
anbi2i |
|- ( ( W e. T /\ ( X K Y ) e. W ) <-> ( W e. T /\ ( K ` <. X , Y >. ) e. W ) ) |
53 |
50 52
|
sylib |
|- ( ph -> ( W e. T /\ ( K ` <. X , Y >. ) e. W ) ) |
54 |
|
xpss12 |
|- ( ( U C_ ( 0 [,] 1 ) /\ V C_ ( 0 [,] 1 ) ) -> ( U X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
55 |
31 35 54
|
syl2anc |
|- ( ph -> ( U X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
56 |
|
snidg |
|- ( m e. U -> m e. { m } ) |
57 |
56
|
ad2antrl |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> m e. { m } ) |
58 |
|
simprr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> n e. V ) |
59 |
|
ovres |
|- ( ( m e. { m } /\ n e. V ) -> ( m ( K |` ( { m } X. V ) ) n ) = ( m K n ) ) |
60 |
57 58 59
|
syl2anc |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( m ( K |` ( { m } X. V ) ) n ) = ( m K n ) ) |
61 |
|
eqid |
|- U. ( ( II tX II ) |`t ( { m } X. V ) ) = U. ( ( II tX II ) |`t ( { m } X. V ) ) |
62 |
21
|
a1i |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> II e. Top ) |
63 |
|
snex |
|- { m } e. _V |
64 |
63
|
a1i |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> { m } e. _V ) |
65 |
12
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> V e. II ) |
66 |
|
txrest |
|- ( ( ( II e. Top /\ II e. Top ) /\ ( { m } e. _V /\ V e. II ) ) -> ( ( II tX II ) |`t ( { m } X. V ) ) = ( ( II |`t { m } ) tX ( II |`t V ) ) ) |
67 |
62 62 64 65 66
|
syl22anc |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( II tX II ) |`t ( { m } X. V ) ) = ( ( II |`t { m } ) tX ( II |`t V ) ) ) |
68 |
|
iitopon |
|- II e. ( TopOn ` ( 0 [,] 1 ) ) |
69 |
31
|
sselda |
|- ( ( ph /\ m e. U ) -> m e. ( 0 [,] 1 ) ) |
70 |
69
|
adantrr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> m e. ( 0 [,] 1 ) ) |
71 |
|
restsn2 |
|- ( ( II e. ( TopOn ` ( 0 [,] 1 ) ) /\ m e. ( 0 [,] 1 ) ) -> ( II |`t { m } ) = ~P { m } ) |
72 |
68 70 71
|
sylancr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( II |`t { m } ) = ~P { m } ) |
73 |
|
pwsn |
|- ~P { m } = { (/) , { m } } |
74 |
|
indisconn |
|- { (/) , { m } } e. Conn |
75 |
73 74
|
eqeltri |
|- ~P { m } e. Conn |
76 |
72 75
|
eqeltrdi |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( II |`t { m } ) e. Conn ) |
77 |
14
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( II |`t V ) e. Conn ) |
78 |
|
txconn |
|- ( ( ( II |`t { m } ) e. Conn /\ ( II |`t V ) e. Conn ) -> ( ( II |`t { m } ) tX ( II |`t V ) ) e. Conn ) |
79 |
76 77 78
|
syl2anc |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( II |`t { m } ) tX ( II |`t V ) ) e. Conn ) |
80 |
67 79
|
eqeltrd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( II tX II ) |`t ( { m } X. V ) ) e. Conn ) |
81 |
1 2 3 4 5 6 7
|
cvmlift2lem6 |
|- ( ( ph /\ m e. ( 0 [,] 1 ) ) -> ( K |` ( { m } X. ( 0 [,] 1 ) ) ) e. ( ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) Cn C ) ) |
82 |
70 81
|
syldan |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( K |` ( { m } X. ( 0 [,] 1 ) ) ) e. ( ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) Cn C ) ) |
83 |
35
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> V C_ ( 0 [,] 1 ) ) |
84 |
|
xpss2 |
|- ( V C_ ( 0 [,] 1 ) -> ( { m } X. V ) C_ ( { m } X. ( 0 [,] 1 ) ) ) |
85 |
83 84
|
syl |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( { m } X. V ) C_ ( { m } X. ( 0 [,] 1 ) ) ) |
86 |
70
|
snssd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> { m } C_ ( 0 [,] 1 ) ) |
87 |
|
xpss1 |
|- ( { m } C_ ( 0 [,] 1 ) -> ( { m } X. ( 0 [,] 1 ) ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
88 |
86 87
|
syl |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( { m } X. ( 0 [,] 1 ) ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
89 |
23
|
restuni |
|- ( ( ( II tX II ) e. Top /\ ( { m } X. ( 0 [,] 1 ) ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( { m } X. ( 0 [,] 1 ) ) = U. ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) ) |
90 |
27 88 89
|
sylancr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( { m } X. ( 0 [,] 1 ) ) = U. ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) ) |
91 |
85 90
|
sseqtrd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( { m } X. V ) C_ U. ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) ) |
92 |
|
eqid |
|- U. ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) = U. ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) |
93 |
92
|
cnrest |
|- ( ( ( K |` ( { m } X. ( 0 [,] 1 ) ) ) e. ( ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) Cn C ) /\ ( { m } X. V ) C_ U. ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) ) -> ( ( K |` ( { m } X. ( 0 [,] 1 ) ) ) |` ( { m } X. V ) ) e. ( ( ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) |`t ( { m } X. V ) ) Cn C ) ) |
94 |
82 91 93
|
syl2anc |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( K |` ( { m } X. ( 0 [,] 1 ) ) ) |` ( { m } X. V ) ) e. ( ( ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) |`t ( { m } X. V ) ) Cn C ) ) |
95 |
85
|
resabs1d |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( K |` ( { m } X. ( 0 [,] 1 ) ) ) |` ( { m } X. V ) ) = ( K |` ( { m } X. V ) ) ) |
96 |
27
|
a1i |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( II tX II ) e. Top ) |
97 |
|
ovex |
|- ( 0 [,] 1 ) e. _V |
98 |
63 97
|
xpex |
|- ( { m } X. ( 0 [,] 1 ) ) e. _V |
99 |
98
|
a1i |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( { m } X. ( 0 [,] 1 ) ) e. _V ) |
100 |
|
restabs |
|- ( ( ( II tX II ) e. Top /\ ( { m } X. V ) C_ ( { m } X. ( 0 [,] 1 ) ) /\ ( { m } X. ( 0 [,] 1 ) ) e. _V ) -> ( ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) |`t ( { m } X. V ) ) = ( ( II tX II ) |`t ( { m } X. V ) ) ) |
101 |
96 85 99 100
|
syl3anc |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) |`t ( { m } X. V ) ) = ( ( II tX II ) |`t ( { m } X. V ) ) ) |
102 |
101
|
oveq1d |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( ( ( II tX II ) |`t ( { m } X. ( 0 [,] 1 ) ) ) |`t ( { m } X. V ) ) Cn C ) = ( ( ( II tX II ) |`t ( { m } X. V ) ) Cn C ) ) |
103 |
94 95 102
|
3eltr3d |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( K |` ( { m } X. V ) ) e. ( ( ( II tX II ) |`t ( { m } X. V ) ) Cn C ) ) |
104 |
|
cvmtop1 |
|- ( F e. ( C CovMap J ) -> C e. Top ) |
105 |
2 104
|
syl |
|- ( ph -> C e. Top ) |
106 |
105
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> C e. Top ) |
107 |
1
|
toptopon |
|- ( C e. Top <-> C e. ( TopOn ` B ) ) |
108 |
106 107
|
sylib |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> C e. ( TopOn ` B ) ) |
109 |
|
df-ima |
|- ( K " ( { m } X. V ) ) = ran ( K |` ( { m } X. V ) ) |
110 |
|
simprl |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> m e. U ) |
111 |
110
|
snssd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> { m } C_ U ) |
112 |
|
xpss1 |
|- ( { m } C_ U -> ( { m } X. V ) C_ ( U X. V ) ) |
113 |
|
imass2 |
|- ( ( { m } X. V ) C_ ( U X. V ) -> ( K " ( { m } X. V ) ) C_ ( K " ( U X. V ) ) ) |
114 |
111 112 113
|
3syl |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( K " ( { m } X. V ) ) C_ ( K " ( U X. V ) ) ) |
115 |
|
imaco |
|- ( ( `' K o. `' F ) " M ) = ( `' K " ( `' F " M ) ) |
116 |
|
cnvco |
|- `' ( F o. K ) = ( `' K o. `' F ) |
117 |
25
|
cnveqd |
|- ( ph -> `' ( F o. K ) = `' G ) |
118 |
116 117
|
eqtr3id |
|- ( ph -> ( `' K o. `' F ) = `' G ) |
119 |
118
|
imaeq1d |
|- ( ph -> ( ( `' K o. `' F ) " M ) = ( `' G " M ) ) |
120 |
115 119
|
eqtr3id |
|- ( ph -> ( `' K " ( `' F " M ) ) = ( `' G " M ) ) |
121 |
17 120
|
sseqtrrd |
|- ( ph -> ( U X. V ) C_ ( `' K " ( `' F " M ) ) ) |
122 |
24
|
ffund |
|- ( ph -> Fun K ) |
123 |
24
|
fdmd |
|- ( ph -> dom K = ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
124 |
55 123
|
sseqtrrd |
|- ( ph -> ( U X. V ) C_ dom K ) |
125 |
|
funimass3 |
|- ( ( Fun K /\ ( U X. V ) C_ dom K ) -> ( ( K " ( U X. V ) ) C_ ( `' F " M ) <-> ( U X. V ) C_ ( `' K " ( `' F " M ) ) ) ) |
126 |
122 124 125
|
syl2anc |
|- ( ph -> ( ( K " ( U X. V ) ) C_ ( `' F " M ) <-> ( U X. V ) C_ ( `' K " ( `' F " M ) ) ) ) |
127 |
121 126
|
mpbird |
|- ( ph -> ( K " ( U X. V ) ) C_ ( `' F " M ) ) |
128 |
127
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( K " ( U X. V ) ) C_ ( `' F " M ) ) |
129 |
114 128
|
sstrd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( K " ( { m } X. V ) ) C_ ( `' F " M ) ) |
130 |
109 129
|
eqsstrrid |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ran ( K |` ( { m } X. V ) ) C_ ( `' F " M ) ) |
131 |
|
cnvimass |
|- ( `' F " M ) C_ dom F |
132 |
|
cvmcn |
|- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) ) |
133 |
2 132
|
syl |
|- ( ph -> F e. ( C Cn J ) ) |
134 |
|
eqid |
|- U. J = U. J |
135 |
1 134
|
cnf |
|- ( F e. ( C Cn J ) -> F : B --> U. J ) |
136 |
|
fdm |
|- ( F : B --> U. J -> dom F = B ) |
137 |
133 135 136
|
3syl |
|- ( ph -> dom F = B ) |
138 |
131 137
|
sseqtrid |
|- ( ph -> ( `' F " M ) C_ B ) |
139 |
138
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( `' F " M ) C_ B ) |
140 |
|
cnrest2 |
|- ( ( C e. ( TopOn ` B ) /\ ran ( K |` ( { m } X. V ) ) C_ ( `' F " M ) /\ ( `' F " M ) C_ B ) -> ( ( K |` ( { m } X. V ) ) e. ( ( ( II tX II ) |`t ( { m } X. V ) ) Cn C ) <-> ( K |` ( { m } X. V ) ) e. ( ( ( II tX II ) |`t ( { m } X. V ) ) Cn ( C |`t ( `' F " M ) ) ) ) ) |
141 |
108 130 139 140
|
syl3anc |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( K |` ( { m } X. V ) ) e. ( ( ( II tX II ) |`t ( { m } X. V ) ) Cn C ) <-> ( K |` ( { m } X. V ) ) e. ( ( ( II tX II ) |`t ( { m } X. V ) ) Cn ( C |`t ( `' F " M ) ) ) ) ) |
142 |
103 141
|
mpbid |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( K |` ( { m } X. V ) ) e. ( ( ( II tX II ) |`t ( { m } X. V ) ) Cn ( C |`t ( `' F " M ) ) ) ) |
143 |
8
|
cvmsss |
|- ( T e. ( S ` M ) -> T C_ C ) |
144 |
10 143
|
syl |
|- ( ph -> T C_ C ) |
145 |
50
|
simpld |
|- ( ph -> W e. T ) |
146 |
144 145
|
sseldd |
|- ( ph -> W e. C ) |
147 |
|
elssuni |
|- ( W e. T -> W C_ U. T ) |
148 |
145 147
|
syl |
|- ( ph -> W C_ U. T ) |
149 |
8
|
cvmsuni |
|- ( T e. ( S ` M ) -> U. T = ( `' F " M ) ) |
150 |
10 149
|
syl |
|- ( ph -> U. T = ( `' F " M ) ) |
151 |
148 150
|
sseqtrd |
|- ( ph -> W C_ ( `' F " M ) ) |
152 |
8
|
cvmsrcl |
|- ( T e. ( S ` M ) -> M e. J ) |
153 |
10 152
|
syl |
|- ( ph -> M e. J ) |
154 |
|
cnima |
|- ( ( F e. ( C Cn J ) /\ M e. J ) -> ( `' F " M ) e. C ) |
155 |
133 153 154
|
syl2anc |
|- ( ph -> ( `' F " M ) e. C ) |
156 |
|
restopn2 |
|- ( ( C e. Top /\ ( `' F " M ) e. C ) -> ( W e. ( C |`t ( `' F " M ) ) <-> ( W e. C /\ W C_ ( `' F " M ) ) ) ) |
157 |
105 155 156
|
syl2anc |
|- ( ph -> ( W e. ( C |`t ( `' F " M ) ) <-> ( W e. C /\ W C_ ( `' F " M ) ) ) ) |
158 |
146 151 157
|
mpbir2and |
|- ( ph -> W e. ( C |`t ( `' F " M ) ) ) |
159 |
158
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> W e. ( C |`t ( `' F " M ) ) ) |
160 |
8
|
cvmscld |
|- ( ( F e. ( C CovMap J ) /\ T e. ( S ` M ) /\ W e. T ) -> W e. ( Clsd ` ( C |`t ( `' F " M ) ) ) ) |
161 |
2 10 145 160
|
syl3anc |
|- ( ph -> W e. ( Clsd ` ( C |`t ( `' F " M ) ) ) ) |
162 |
161
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> W e. ( Clsd ` ( C |`t ( `' F " M ) ) ) ) |
163 |
18
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> Z e. V ) |
164 |
|
opelxpi |
|- ( ( m e. { m } /\ Z e. V ) -> <. m , Z >. e. ( { m } X. V ) ) |
165 |
57 163 164
|
syl2anc |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> <. m , Z >. e. ( { m } X. V ) ) |
166 |
85 88
|
sstrd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( { m } X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
167 |
23
|
restuni |
|- ( ( ( II tX II ) e. Top /\ ( { m } X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( { m } X. V ) = U. ( ( II tX II ) |`t ( { m } X. V ) ) ) |
168 |
27 166 167
|
sylancr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( { m } X. V ) = U. ( ( II tX II ) |`t ( { m } X. V ) ) ) |
169 |
165 168
|
eleqtrd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> <. m , Z >. e. U. ( ( II tX II ) |`t ( { m } X. V ) ) ) |
170 |
|
df-ov |
|- ( m ( K |` ( { m } X. V ) ) Z ) = ( ( K |` ( { m } X. V ) ) ` <. m , Z >. ) |
171 |
|
ovres |
|- ( ( m e. { m } /\ Z e. V ) -> ( m ( K |` ( { m } X. V ) ) Z ) = ( m K Z ) ) |
172 |
57 163 171
|
syl2anc |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( m ( K |` ( { m } X. V ) ) Z ) = ( m K Z ) ) |
173 |
|
snidg |
|- ( Z e. V -> Z e. { Z } ) |
174 |
18 173
|
syl |
|- ( ph -> Z e. { Z } ) |
175 |
174
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> Z e. { Z } ) |
176 |
|
ovres |
|- ( ( m e. U /\ Z e. { Z } ) -> ( m ( K |` ( U X. { Z } ) ) Z ) = ( m K Z ) ) |
177 |
110 175 176
|
syl2anc |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( m ( K |` ( U X. { Z } ) ) Z ) = ( m K Z ) ) |
178 |
172 177
|
eqtr4d |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( m ( K |` ( { m } X. V ) ) Z ) = ( m ( K |` ( U X. { Z } ) ) Z ) ) |
179 |
170 178
|
eqtr3id |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( K |` ( { m } X. V ) ) ` <. m , Z >. ) = ( m ( K |` ( U X. { Z } ) ) Z ) ) |
180 |
|
eqid |
|- U. ( ( II tX II ) |`t ( U X. { Z } ) ) = U. ( ( II tX II ) |`t ( U X. { Z } ) ) |
181 |
21
|
a1i |
|- ( ph -> II e. Top ) |
182 |
|
snex |
|- { Z } e. _V |
183 |
182
|
a1i |
|- ( ph -> { Z } e. _V ) |
184 |
|
txrest |
|- ( ( ( II e. Top /\ II e. Top ) /\ ( U e. II /\ { Z } e. _V ) ) -> ( ( II tX II ) |`t ( U X. { Z } ) ) = ( ( II |`t U ) tX ( II |`t { Z } ) ) ) |
185 |
181 181 11 183 184
|
syl22anc |
|- ( ph -> ( ( II tX II ) |`t ( U X. { Z } ) ) = ( ( II |`t U ) tX ( II |`t { Z } ) ) ) |
186 |
35 18
|
sseldd |
|- ( ph -> Z e. ( 0 [,] 1 ) ) |
187 |
|
restsn2 |
|- ( ( II e. ( TopOn ` ( 0 [,] 1 ) ) /\ Z e. ( 0 [,] 1 ) ) -> ( II |`t { Z } ) = ~P { Z } ) |
188 |
68 186 187
|
sylancr |
|- ( ph -> ( II |`t { Z } ) = ~P { Z } ) |
189 |
|
pwsn |
|- ~P { Z } = { (/) , { Z } } |
190 |
|
indisconn |
|- { (/) , { Z } } e. Conn |
191 |
189 190
|
eqeltri |
|- ~P { Z } e. Conn |
192 |
188 191
|
eqeltrdi |
|- ( ph -> ( II |`t { Z } ) e. Conn ) |
193 |
|
txconn |
|- ( ( ( II |`t U ) e. Conn /\ ( II |`t { Z } ) e. Conn ) -> ( ( II |`t U ) tX ( II |`t { Z } ) ) e. Conn ) |
194 |
13 192 193
|
syl2anc |
|- ( ph -> ( ( II |`t U ) tX ( II |`t { Z } ) ) e. Conn ) |
195 |
185 194
|
eqeltrd |
|- ( ph -> ( ( II tX II ) |`t ( U X. { Z } ) ) e. Conn ) |
196 |
105 107
|
sylib |
|- ( ph -> C e. ( TopOn ` B ) ) |
197 |
|
df-ima |
|- ( K " ( U X. { Z } ) ) = ran ( K |` ( U X. { Z } ) ) |
198 |
18
|
snssd |
|- ( ph -> { Z } C_ V ) |
199 |
|
xpss2 |
|- ( { Z } C_ V -> ( U X. { Z } ) C_ ( U X. V ) ) |
200 |
|
imass2 |
|- ( ( U X. { Z } ) C_ ( U X. V ) -> ( K " ( U X. { Z } ) ) C_ ( K " ( U X. V ) ) ) |
201 |
198 199 200
|
3syl |
|- ( ph -> ( K " ( U X. { Z } ) ) C_ ( K " ( U X. V ) ) ) |
202 |
201 127
|
sstrd |
|- ( ph -> ( K " ( U X. { Z } ) ) C_ ( `' F " M ) ) |
203 |
197 202
|
eqsstrrid |
|- ( ph -> ran ( K |` ( U X. { Z } ) ) C_ ( `' F " M ) ) |
204 |
|
cnrest2 |
|- ( ( C e. ( TopOn ` B ) /\ ran ( K |` ( U X. { Z } ) ) C_ ( `' F " M ) /\ ( `' F " M ) C_ B ) -> ( ( K |` ( U X. { Z } ) ) e. ( ( ( II tX II ) |`t ( U X. { Z } ) ) Cn C ) <-> ( K |` ( U X. { Z } ) ) e. ( ( ( II tX II ) |`t ( U X. { Z } ) ) Cn ( C |`t ( `' F " M ) ) ) ) ) |
205 |
196 203 138 204
|
syl3anc |
|- ( ph -> ( ( K |` ( U X. { Z } ) ) e. ( ( ( II tX II ) |`t ( U X. { Z } ) ) Cn C ) <-> ( K |` ( U X. { Z } ) ) e. ( ( ( II tX II ) |`t ( U X. { Z } ) ) Cn ( C |`t ( `' F " M ) ) ) ) ) |
206 |
19 205
|
mpbid |
|- ( ph -> ( K |` ( U X. { Z } ) ) e. ( ( ( II tX II ) |`t ( U X. { Z } ) ) Cn ( C |`t ( `' F " M ) ) ) ) |
207 |
|
opelxpi |
|- ( ( X e. U /\ Z e. { Z } ) -> <. X , Z >. e. ( U X. { Z } ) ) |
208 |
15 174 207
|
syl2anc |
|- ( ph -> <. X , Z >. e. ( U X. { Z } ) ) |
209 |
186
|
snssd |
|- ( ph -> { Z } C_ ( 0 [,] 1 ) ) |
210 |
|
xpss12 |
|- ( ( U C_ ( 0 [,] 1 ) /\ { Z } C_ ( 0 [,] 1 ) ) -> ( U X. { Z } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
211 |
31 209 210
|
syl2anc |
|- ( ph -> ( U X. { Z } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
212 |
23
|
restuni |
|- ( ( ( II tX II ) e. Top /\ ( U X. { Z } ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( U X. { Z } ) = U. ( ( II tX II ) |`t ( U X. { Z } ) ) ) |
213 |
27 211 212
|
sylancr |
|- ( ph -> ( U X. { Z } ) = U. ( ( II tX II ) |`t ( U X. { Z } ) ) ) |
214 |
208 213
|
eleqtrd |
|- ( ph -> <. X , Z >. e. U. ( ( II tX II ) |`t ( U X. { Z } ) ) ) |
215 |
|
df-ov |
|- ( X ( K |` ( U X. { Z } ) ) Z ) = ( ( K |` ( U X. { Z } ) ) ` <. X , Z >. ) |
216 |
|
ovres |
|- ( ( X e. U /\ Z e. { Z } ) -> ( X ( K |` ( U X. { Z } ) ) Z ) = ( X K Z ) ) |
217 |
15 174 216
|
syl2anc |
|- ( ph -> ( X ( K |` ( U X. { Z } ) ) Z ) = ( X K Z ) ) |
218 |
|
snidg |
|- ( X e. U -> X e. { X } ) |
219 |
15 218
|
syl |
|- ( ph -> X e. { X } ) |
220 |
|
ovres |
|- ( ( X e. { X } /\ Z e. V ) -> ( X ( K |` ( { X } X. V ) ) Z ) = ( X K Z ) ) |
221 |
219 18 220
|
syl2anc |
|- ( ph -> ( X ( K |` ( { X } X. V ) ) Z ) = ( X K Z ) ) |
222 |
217 221
|
eqtr4d |
|- ( ph -> ( X ( K |` ( U X. { Z } ) ) Z ) = ( X ( K |` ( { X } X. V ) ) Z ) ) |
223 |
215 222
|
eqtr3id |
|- ( ph -> ( ( K |` ( U X. { Z } ) ) ` <. X , Z >. ) = ( X ( K |` ( { X } X. V ) ) Z ) ) |
224 |
|
eqid |
|- U. ( ( II tX II ) |`t ( { X } X. V ) ) = U. ( ( II tX II ) |`t ( { X } X. V ) ) |
225 |
|
snex |
|- { X } e. _V |
226 |
225
|
a1i |
|- ( ph -> { X } e. _V ) |
227 |
|
txrest |
|- ( ( ( II e. Top /\ II e. Top ) /\ ( { X } e. _V /\ V e. II ) ) -> ( ( II tX II ) |`t ( { X } X. V ) ) = ( ( II |`t { X } ) tX ( II |`t V ) ) ) |
228 |
181 181 226 12 227
|
syl22anc |
|- ( ph -> ( ( II tX II ) |`t ( { X } X. V ) ) = ( ( II |`t { X } ) tX ( II |`t V ) ) ) |
229 |
|
restsn2 |
|- ( ( II e. ( TopOn ` ( 0 [,] 1 ) ) /\ X e. ( 0 [,] 1 ) ) -> ( II |`t { X } ) = ~P { X } ) |
230 |
68 32 229
|
sylancr |
|- ( ph -> ( II |`t { X } ) = ~P { X } ) |
231 |
|
pwsn |
|- ~P { X } = { (/) , { X } } |
232 |
|
indisconn |
|- { (/) , { X } } e. Conn |
233 |
231 232
|
eqeltri |
|- ~P { X } e. Conn |
234 |
230 233
|
eqeltrdi |
|- ( ph -> ( II |`t { X } ) e. Conn ) |
235 |
|
txconn |
|- ( ( ( II |`t { X } ) e. Conn /\ ( II |`t V ) e. Conn ) -> ( ( II |`t { X } ) tX ( II |`t V ) ) e. Conn ) |
236 |
234 14 235
|
syl2anc |
|- ( ph -> ( ( II |`t { X } ) tX ( II |`t V ) ) e. Conn ) |
237 |
228 236
|
eqeltrd |
|- ( ph -> ( ( II tX II ) |`t ( { X } X. V ) ) e. Conn ) |
238 |
1 2 3 4 5 6 7
|
cvmlift2lem6 |
|- ( ( ph /\ X e. ( 0 [,] 1 ) ) -> ( K |` ( { X } X. ( 0 [,] 1 ) ) ) e. ( ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) Cn C ) ) |
239 |
32 238
|
mpdan |
|- ( ph -> ( K |` ( { X } X. ( 0 [,] 1 ) ) ) e. ( ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) Cn C ) ) |
240 |
|
xpss2 |
|- ( V C_ ( 0 [,] 1 ) -> ( { X } X. V ) C_ ( { X } X. ( 0 [,] 1 ) ) ) |
241 |
12 34 240
|
3syl |
|- ( ph -> ( { X } X. V ) C_ ( { X } X. ( 0 [,] 1 ) ) ) |
242 |
32
|
snssd |
|- ( ph -> { X } C_ ( 0 [,] 1 ) ) |
243 |
|
xpss1 |
|- ( { X } C_ ( 0 [,] 1 ) -> ( { X } X. ( 0 [,] 1 ) ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
244 |
242 243
|
syl |
|- ( ph -> ( { X } X. ( 0 [,] 1 ) ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
245 |
23
|
restuni |
|- ( ( ( II tX II ) e. Top /\ ( { X } X. ( 0 [,] 1 ) ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( { X } X. ( 0 [,] 1 ) ) = U. ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) ) |
246 |
27 244 245
|
sylancr |
|- ( ph -> ( { X } X. ( 0 [,] 1 ) ) = U. ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) ) |
247 |
241 246
|
sseqtrd |
|- ( ph -> ( { X } X. V ) C_ U. ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) ) |
248 |
|
eqid |
|- U. ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) = U. ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) |
249 |
248
|
cnrest |
|- ( ( ( K |` ( { X } X. ( 0 [,] 1 ) ) ) e. ( ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) Cn C ) /\ ( { X } X. V ) C_ U. ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) ) -> ( ( K |` ( { X } X. ( 0 [,] 1 ) ) ) |` ( { X } X. V ) ) e. ( ( ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) |`t ( { X } X. V ) ) Cn C ) ) |
250 |
239 247 249
|
syl2anc |
|- ( ph -> ( ( K |` ( { X } X. ( 0 [,] 1 ) ) ) |` ( { X } X. V ) ) e. ( ( ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) |`t ( { X } X. V ) ) Cn C ) ) |
251 |
241
|
resabs1d |
|- ( ph -> ( ( K |` ( { X } X. ( 0 [,] 1 ) ) ) |` ( { X } X. V ) ) = ( K |` ( { X } X. V ) ) ) |
252 |
225 97
|
xpex |
|- ( { X } X. ( 0 [,] 1 ) ) e. _V |
253 |
252
|
a1i |
|- ( ph -> ( { X } X. ( 0 [,] 1 ) ) e. _V ) |
254 |
|
restabs |
|- ( ( ( II tX II ) e. Top /\ ( { X } X. V ) C_ ( { X } X. ( 0 [,] 1 ) ) /\ ( { X } X. ( 0 [,] 1 ) ) e. _V ) -> ( ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) |`t ( { X } X. V ) ) = ( ( II tX II ) |`t ( { X } X. V ) ) ) |
255 |
28 241 253 254
|
syl3anc |
|- ( ph -> ( ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) |`t ( { X } X. V ) ) = ( ( II tX II ) |`t ( { X } X. V ) ) ) |
256 |
255
|
oveq1d |
|- ( ph -> ( ( ( ( II tX II ) |`t ( { X } X. ( 0 [,] 1 ) ) ) |`t ( { X } X. V ) ) Cn C ) = ( ( ( II tX II ) |`t ( { X } X. V ) ) Cn C ) ) |
257 |
250 251 256
|
3eltr3d |
|- ( ph -> ( K |` ( { X } X. V ) ) e. ( ( ( II tX II ) |`t ( { X } X. V ) ) Cn C ) ) |
258 |
|
df-ima |
|- ( K " ( { X } X. V ) ) = ran ( K |` ( { X } X. V ) ) |
259 |
15
|
snssd |
|- ( ph -> { X } C_ U ) |
260 |
|
xpss1 |
|- ( { X } C_ U -> ( { X } X. V ) C_ ( U X. V ) ) |
261 |
|
imass2 |
|- ( ( { X } X. V ) C_ ( U X. V ) -> ( K " ( { X } X. V ) ) C_ ( K " ( U X. V ) ) ) |
262 |
259 260 261
|
3syl |
|- ( ph -> ( K " ( { X } X. V ) ) C_ ( K " ( U X. V ) ) ) |
263 |
262 127
|
sstrd |
|- ( ph -> ( K " ( { X } X. V ) ) C_ ( `' F " M ) ) |
264 |
258 263
|
eqsstrrid |
|- ( ph -> ran ( K |` ( { X } X. V ) ) C_ ( `' F " M ) ) |
265 |
|
cnrest2 |
|- ( ( C e. ( TopOn ` B ) /\ ran ( K |` ( { X } X. V ) ) C_ ( `' F " M ) /\ ( `' F " M ) C_ B ) -> ( ( K |` ( { X } X. V ) ) e. ( ( ( II tX II ) |`t ( { X } X. V ) ) Cn C ) <-> ( K |` ( { X } X. V ) ) e. ( ( ( II tX II ) |`t ( { X } X. V ) ) Cn ( C |`t ( `' F " M ) ) ) ) ) |
266 |
196 264 138 265
|
syl3anc |
|- ( ph -> ( ( K |` ( { X } X. V ) ) e. ( ( ( II tX II ) |`t ( { X } X. V ) ) Cn C ) <-> ( K |` ( { X } X. V ) ) e. ( ( ( II tX II ) |`t ( { X } X. V ) ) Cn ( C |`t ( `' F " M ) ) ) ) ) |
267 |
257 266
|
mpbid |
|- ( ph -> ( K |` ( { X } X. V ) ) e. ( ( ( II tX II ) |`t ( { X } X. V ) ) Cn ( C |`t ( `' F " M ) ) ) ) |
268 |
|
opelxpi |
|- ( ( X e. { X } /\ Y e. V ) -> <. X , Y >. e. ( { X } X. V ) ) |
269 |
219 16 268
|
syl2anc |
|- ( ph -> <. X , Y >. e. ( { X } X. V ) ) |
270 |
259 260
|
syl |
|- ( ph -> ( { X } X. V ) C_ ( U X. V ) ) |
271 |
270 55
|
sstrd |
|- ( ph -> ( { X } X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) |
272 |
23
|
restuni |
|- ( ( ( II tX II ) e. Top /\ ( { X } X. V ) C_ ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( { X } X. V ) = U. ( ( II tX II ) |`t ( { X } X. V ) ) ) |
273 |
27 271 272
|
sylancr |
|- ( ph -> ( { X } X. V ) = U. ( ( II tX II ) |`t ( { X } X. V ) ) ) |
274 |
269 273
|
eleqtrd |
|- ( ph -> <. X , Y >. e. U. ( ( II tX II ) |`t ( { X } X. V ) ) ) |
275 |
|
df-ov |
|- ( X ( K |` ( { X } X. V ) ) Y ) = ( ( K |` ( { X } X. V ) ) ` <. X , Y >. ) |
276 |
|
ovres |
|- ( ( X e. { X } /\ Y e. V ) -> ( X ( K |` ( { X } X. V ) ) Y ) = ( X K Y ) ) |
277 |
219 16 276
|
syl2anc |
|- ( ph -> ( X ( K |` ( { X } X. V ) ) Y ) = ( X K Y ) ) |
278 |
275 277
|
eqtr3id |
|- ( ph -> ( ( K |` ( { X } X. V ) ) ` <. X , Y >. ) = ( X K Y ) ) |
279 |
50
|
simprd |
|- ( ph -> ( X K Y ) e. W ) |
280 |
278 279
|
eqeltrd |
|- ( ph -> ( ( K |` ( { X } X. V ) ) ` <. X , Y >. ) e. W ) |
281 |
224 237 267 158 161 274 280
|
conncn |
|- ( ph -> ( K |` ( { X } X. V ) ) : U. ( ( II tX II ) |`t ( { X } X. V ) ) --> W ) |
282 |
273
|
feq2d |
|- ( ph -> ( ( K |` ( { X } X. V ) ) : ( { X } X. V ) --> W <-> ( K |` ( { X } X. V ) ) : U. ( ( II tX II ) |`t ( { X } X. V ) ) --> W ) ) |
283 |
281 282
|
mpbird |
|- ( ph -> ( K |` ( { X } X. V ) ) : ( { X } X. V ) --> W ) |
284 |
283 219 18
|
fovrnd |
|- ( ph -> ( X ( K |` ( { X } X. V ) ) Z ) e. W ) |
285 |
223 284
|
eqeltrd |
|- ( ph -> ( ( K |` ( U X. { Z } ) ) ` <. X , Z >. ) e. W ) |
286 |
180 195 206 158 161 214 285
|
conncn |
|- ( ph -> ( K |` ( U X. { Z } ) ) : U. ( ( II tX II ) |`t ( U X. { Z } ) ) --> W ) |
287 |
213
|
feq2d |
|- ( ph -> ( ( K |` ( U X. { Z } ) ) : ( U X. { Z } ) --> W <-> ( K |` ( U X. { Z } ) ) : U. ( ( II tX II ) |`t ( U X. { Z } ) ) --> W ) ) |
288 |
286 287
|
mpbird |
|- ( ph -> ( K |` ( U X. { Z } ) ) : ( U X. { Z } ) --> W ) |
289 |
288
|
adantr |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( K |` ( U X. { Z } ) ) : ( U X. { Z } ) --> W ) |
290 |
289 110 175
|
fovrnd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( m ( K |` ( U X. { Z } ) ) Z ) e. W ) |
291 |
179 290
|
eqeltrd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( K |` ( { m } X. V ) ) ` <. m , Z >. ) e. W ) |
292 |
61 80 142 159 162 169 291
|
conncn |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( K |` ( { m } X. V ) ) : U. ( ( II tX II ) |`t ( { m } X. V ) ) --> W ) |
293 |
168
|
feq2d |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( ( K |` ( { m } X. V ) ) : ( { m } X. V ) --> W <-> ( K |` ( { m } X. V ) ) : U. ( ( II tX II ) |`t ( { m } X. V ) ) --> W ) ) |
294 |
292 293
|
mpbird |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( K |` ( { m } X. V ) ) : ( { m } X. V ) --> W ) |
295 |
294 57 58
|
fovrnd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( m ( K |` ( { m } X. V ) ) n ) e. W ) |
296 |
60 295
|
eqeltrrd |
|- ( ( ph /\ ( m e. U /\ n e. V ) ) -> ( m K n ) e. W ) |
297 |
296
|
ralrimivva |
|- ( ph -> A. m e. U A. n e. V ( m K n ) e. W ) |
298 |
|
funimassov |
|- ( ( Fun K /\ ( U X. V ) C_ dom K ) -> ( ( K " ( U X. V ) ) C_ W <-> A. m e. U A. n e. V ( m K n ) e. W ) ) |
299 |
122 124 298
|
syl2anc |
|- ( ph -> ( ( K " ( U X. V ) ) C_ W <-> A. m e. U A. n e. V ( m K n ) e. W ) ) |
300 |
297 299
|
mpbird |
|- ( ph -> ( K " ( U X. V ) ) C_ W ) |
301 |
1 23 8 2 24 26 28 38 10 53 55 300
|
cvmlift2lem9a |
|- ( ph -> ( K |` ( U X. V ) ) e. ( ( ( II tX II ) |`t ( U X. V ) ) Cn C ) ) |