Step |
Hyp |
Ref |
Expression |
1 |
|
heibor.1 |
|- J = ( MetOpen ` D ) |
2 |
|
heibor.3 |
|- K = { u | -. E. v e. ( ~P U i^i Fin ) u C_ U. v } |
3 |
|
heibor.4 |
|- G = { <. y , n >. | ( n e. NN0 /\ y e. ( F ` n ) /\ ( y B n ) e. K ) } |
4 |
|
heibor.5 |
|- B = ( z e. X , m e. NN0 |-> ( z ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) ) |
5 |
|
heibor.6 |
|- ( ph -> D e. ( CMet ` X ) ) |
6 |
|
heibor.7 |
|- ( ph -> F : NN0 --> ( ~P X i^i Fin ) ) |
7 |
|
heibor.8 |
|- ( ph -> A. n e. NN0 X = U_ y e. ( F ` n ) ( y B n ) ) |
8 |
|
nn0ex |
|- NN0 e. _V |
9 |
|
fvex |
|- ( F ` t ) e. _V |
10 |
|
snex |
|- { t } e. _V |
11 |
9 10
|
xpex |
|- ( ( F ` t ) X. { t } ) e. _V |
12 |
8 11
|
iunex |
|- U_ t e. NN0 ( ( F ` t ) X. { t } ) e. _V |
13 |
3
|
relopabiv |
|- Rel G |
14 |
|
1st2nd |
|- ( ( Rel G /\ x e. G ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) |
15 |
13 14
|
mpan |
|- ( x e. G -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) |
16 |
15
|
eleq1d |
|- ( x e. G -> ( x e. G <-> <. ( 1st ` x ) , ( 2nd ` x ) >. e. G ) ) |
17 |
|
df-br |
|- ( ( 1st ` x ) G ( 2nd ` x ) <-> <. ( 1st ` x ) , ( 2nd ` x ) >. e. G ) |
18 |
16 17
|
bitr4di |
|- ( x e. G -> ( x e. G <-> ( 1st ` x ) G ( 2nd ` x ) ) ) |
19 |
|
fvex |
|- ( 1st ` x ) e. _V |
20 |
|
fvex |
|- ( 2nd ` x ) e. _V |
21 |
1 2 3 19 20
|
heiborlem2 |
|- ( ( 1st ` x ) G ( 2nd ` x ) <-> ( ( 2nd ` x ) e. NN0 /\ ( 1st ` x ) e. ( F ` ( 2nd ` x ) ) /\ ( ( 1st ` x ) B ( 2nd ` x ) ) e. K ) ) |
22 |
18 21
|
bitrdi |
|- ( x e. G -> ( x e. G <-> ( ( 2nd ` x ) e. NN0 /\ ( 1st ` x ) e. ( F ` ( 2nd ` x ) ) /\ ( ( 1st ` x ) B ( 2nd ` x ) ) e. K ) ) ) |
23 |
22
|
ibi |
|- ( x e. G -> ( ( 2nd ` x ) e. NN0 /\ ( 1st ` x ) e. ( F ` ( 2nd ` x ) ) /\ ( ( 1st ` x ) B ( 2nd ` x ) ) e. K ) ) |
24 |
20
|
snid |
|- ( 2nd ` x ) e. { ( 2nd ` x ) } |
25 |
|
opelxp |
|- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( F ` ( 2nd ` x ) ) X. { ( 2nd ` x ) } ) <-> ( ( 1st ` x ) e. ( F ` ( 2nd ` x ) ) /\ ( 2nd ` x ) e. { ( 2nd ` x ) } ) ) |
26 |
24 25
|
mpbiran2 |
|- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( F ` ( 2nd ` x ) ) X. { ( 2nd ` x ) } ) <-> ( 1st ` x ) e. ( F ` ( 2nd ` x ) ) ) |
27 |
|
fveq2 |
|- ( t = ( 2nd ` x ) -> ( F ` t ) = ( F ` ( 2nd ` x ) ) ) |
28 |
|
sneq |
|- ( t = ( 2nd ` x ) -> { t } = { ( 2nd ` x ) } ) |
29 |
27 28
|
xpeq12d |
|- ( t = ( 2nd ` x ) -> ( ( F ` t ) X. { t } ) = ( ( F ` ( 2nd ` x ) ) X. { ( 2nd ` x ) } ) ) |
30 |
29
|
eleq2d |
|- ( t = ( 2nd ` x ) -> ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( F ` t ) X. { t } ) <-> <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( F ` ( 2nd ` x ) ) X. { ( 2nd ` x ) } ) ) ) |
31 |
30
|
rspcev |
|- ( ( ( 2nd ` x ) e. NN0 /\ <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( F ` ( 2nd ` x ) ) X. { ( 2nd ` x ) } ) ) -> E. t e. NN0 <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( F ` t ) X. { t } ) ) |
32 |
26 31
|
sylan2br |
|- ( ( ( 2nd ` x ) e. NN0 /\ ( 1st ` x ) e. ( F ` ( 2nd ` x ) ) ) -> E. t e. NN0 <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( F ` t ) X. { t } ) ) |
33 |
|
eliun |
|- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. U_ t e. NN0 ( ( F ` t ) X. { t } ) <-> E. t e. NN0 <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( ( F ` t ) X. { t } ) ) |
34 |
32 33
|
sylibr |
|- ( ( ( 2nd ` x ) e. NN0 /\ ( 1st ` x ) e. ( F ` ( 2nd ` x ) ) ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. U_ t e. NN0 ( ( F ` t ) X. { t } ) ) |
35 |
34
|
3adant3 |
|- ( ( ( 2nd ` x ) e. NN0 /\ ( 1st ` x ) e. ( F ` ( 2nd ` x ) ) /\ ( ( 1st ` x ) B ( 2nd ` x ) ) e. K ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. U_ t e. NN0 ( ( F ` t ) X. { t } ) ) |
36 |
23 35
|
syl |
|- ( x e. G -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. U_ t e. NN0 ( ( F ` t ) X. { t } ) ) |
37 |
15 36
|
eqeltrd |
|- ( x e. G -> x e. U_ t e. NN0 ( ( F ` t ) X. { t } ) ) |
38 |
37
|
ssriv |
|- G C_ U_ t e. NN0 ( ( F ` t ) X. { t } ) |
39 |
|
ssdomg |
|- ( U_ t e. NN0 ( ( F ` t ) X. { t } ) e. _V -> ( G C_ U_ t e. NN0 ( ( F ` t ) X. { t } ) -> G ~<_ U_ t e. NN0 ( ( F ` t ) X. { t } ) ) ) |
40 |
12 38 39
|
mp2 |
|- G ~<_ U_ t e. NN0 ( ( F ` t ) X. { t } ) |
41 |
|
nn0ennn |
|- NN0 ~~ NN |
42 |
|
nnenom |
|- NN ~~ _om |
43 |
41 42
|
entri |
|- NN0 ~~ _om |
44 |
|
endom |
|- ( NN0 ~~ _om -> NN0 ~<_ _om ) |
45 |
43 44
|
ax-mp |
|- NN0 ~<_ _om |
46 |
|
vex |
|- t e. _V |
47 |
9 46
|
xpsnen |
|- ( ( F ` t ) X. { t } ) ~~ ( F ` t ) |
48 |
|
inss2 |
|- ( ~P X i^i Fin ) C_ Fin |
49 |
6
|
ffvelrnda |
|- ( ( ph /\ t e. NN0 ) -> ( F ` t ) e. ( ~P X i^i Fin ) ) |
50 |
48 49
|
sseldi |
|- ( ( ph /\ t e. NN0 ) -> ( F ` t ) e. Fin ) |
51 |
|
isfinite |
|- ( ( F ` t ) e. Fin <-> ( F ` t ) ~< _om ) |
52 |
|
sdomdom |
|- ( ( F ` t ) ~< _om -> ( F ` t ) ~<_ _om ) |
53 |
51 52
|
sylbi |
|- ( ( F ` t ) e. Fin -> ( F ` t ) ~<_ _om ) |
54 |
50 53
|
syl |
|- ( ( ph /\ t e. NN0 ) -> ( F ` t ) ~<_ _om ) |
55 |
|
endomtr |
|- ( ( ( ( F ` t ) X. { t } ) ~~ ( F ` t ) /\ ( F ` t ) ~<_ _om ) -> ( ( F ` t ) X. { t } ) ~<_ _om ) |
56 |
47 54 55
|
sylancr |
|- ( ( ph /\ t e. NN0 ) -> ( ( F ` t ) X. { t } ) ~<_ _om ) |
57 |
56
|
ralrimiva |
|- ( ph -> A. t e. NN0 ( ( F ` t ) X. { t } ) ~<_ _om ) |
58 |
|
iunctb |
|- ( ( NN0 ~<_ _om /\ A. t e. NN0 ( ( F ` t ) X. { t } ) ~<_ _om ) -> U_ t e. NN0 ( ( F ` t ) X. { t } ) ~<_ _om ) |
59 |
45 57 58
|
sylancr |
|- ( ph -> U_ t e. NN0 ( ( F ` t ) X. { t } ) ~<_ _om ) |
60 |
|
domtr |
|- ( ( G ~<_ U_ t e. NN0 ( ( F ` t ) X. { t } ) /\ U_ t e. NN0 ( ( F ` t ) X. { t } ) ~<_ _om ) -> G ~<_ _om ) |
61 |
40 59 60
|
sylancr |
|- ( ph -> G ~<_ _om ) |
62 |
23
|
simp1d |
|- ( x e. G -> ( 2nd ` x ) e. NN0 ) |
63 |
|
peano2nn0 |
|- ( ( 2nd ` x ) e. NN0 -> ( ( 2nd ` x ) + 1 ) e. NN0 ) |
64 |
62 63
|
syl |
|- ( x e. G -> ( ( 2nd ` x ) + 1 ) e. NN0 ) |
65 |
|
ffvelrn |
|- ( ( F : NN0 --> ( ~P X i^i Fin ) /\ ( ( 2nd ` x ) + 1 ) e. NN0 ) -> ( F ` ( ( 2nd ` x ) + 1 ) ) e. ( ~P X i^i Fin ) ) |
66 |
6 64 65
|
syl2an |
|- ( ( ph /\ x e. G ) -> ( F ` ( ( 2nd ` x ) + 1 ) ) e. ( ~P X i^i Fin ) ) |
67 |
48 66
|
sseldi |
|- ( ( ph /\ x e. G ) -> ( F ` ( ( 2nd ` x ) + 1 ) ) e. Fin ) |
68 |
|
iunin2 |
|- U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) = ( ( B ` x ) i^i U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B ( ( 2nd ` x ) + 1 ) ) ) |
69 |
|
oveq1 |
|- ( y = t -> ( y B n ) = ( t B n ) ) |
70 |
69
|
cbviunv |
|- U_ y e. ( F ` n ) ( y B n ) = U_ t e. ( F ` n ) ( t B n ) |
71 |
|
fveq2 |
|- ( n = ( ( 2nd ` x ) + 1 ) -> ( F ` n ) = ( F ` ( ( 2nd ` x ) + 1 ) ) ) |
72 |
71
|
iuneq1d |
|- ( n = ( ( 2nd ` x ) + 1 ) -> U_ t e. ( F ` n ) ( t B n ) = U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B n ) ) |
73 |
70 72
|
syl5eq |
|- ( n = ( ( 2nd ` x ) + 1 ) -> U_ y e. ( F ` n ) ( y B n ) = U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B n ) ) |
74 |
|
oveq2 |
|- ( n = ( ( 2nd ` x ) + 1 ) -> ( t B n ) = ( t B ( ( 2nd ` x ) + 1 ) ) ) |
75 |
74
|
iuneq2d |
|- ( n = ( ( 2nd ` x ) + 1 ) -> U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B n ) = U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B ( ( 2nd ` x ) + 1 ) ) ) |
76 |
73 75
|
eqtrd |
|- ( n = ( ( 2nd ` x ) + 1 ) -> U_ y e. ( F ` n ) ( y B n ) = U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B ( ( 2nd ` x ) + 1 ) ) ) |
77 |
76
|
eqeq2d |
|- ( n = ( ( 2nd ` x ) + 1 ) -> ( X = U_ y e. ( F ` n ) ( y B n ) <-> X = U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B ( ( 2nd ` x ) + 1 ) ) ) ) |
78 |
77
|
rspccva |
|- ( ( A. n e. NN0 X = U_ y e. ( F ` n ) ( y B n ) /\ ( ( 2nd ` x ) + 1 ) e. NN0 ) -> X = U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B ( ( 2nd ` x ) + 1 ) ) ) |
79 |
7 64 78
|
syl2an |
|- ( ( ph /\ x e. G ) -> X = U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B ( ( 2nd ` x ) + 1 ) ) ) |
80 |
79
|
ineq2d |
|- ( ( ph /\ x e. G ) -> ( ( B ` x ) i^i X ) = ( ( B ` x ) i^i U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B ( ( 2nd ` x ) + 1 ) ) ) ) |
81 |
15
|
fveq2d |
|- ( x e. G -> ( B ` x ) = ( B ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) ) |
82 |
|
df-ov |
|- ( ( 1st ` x ) B ( 2nd ` x ) ) = ( B ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) |
83 |
81 82
|
eqtr4di |
|- ( x e. G -> ( B ` x ) = ( ( 1st ` x ) B ( 2nd ` x ) ) ) |
84 |
83
|
adantl |
|- ( ( ph /\ x e. G ) -> ( B ` x ) = ( ( 1st ` x ) B ( 2nd ` x ) ) ) |
85 |
|
inss1 |
|- ( ~P X i^i Fin ) C_ ~P X |
86 |
|
ffvelrn |
|- ( ( F : NN0 --> ( ~P X i^i Fin ) /\ ( 2nd ` x ) e. NN0 ) -> ( F ` ( 2nd ` x ) ) e. ( ~P X i^i Fin ) ) |
87 |
6 62 86
|
syl2an |
|- ( ( ph /\ x e. G ) -> ( F ` ( 2nd ` x ) ) e. ( ~P X i^i Fin ) ) |
88 |
85 87
|
sseldi |
|- ( ( ph /\ x e. G ) -> ( F ` ( 2nd ` x ) ) e. ~P X ) |
89 |
88
|
elpwid |
|- ( ( ph /\ x e. G ) -> ( F ` ( 2nd ` x ) ) C_ X ) |
90 |
23
|
simp2d |
|- ( x e. G -> ( 1st ` x ) e. ( F ` ( 2nd ` x ) ) ) |
91 |
90
|
adantl |
|- ( ( ph /\ x e. G ) -> ( 1st ` x ) e. ( F ` ( 2nd ` x ) ) ) |
92 |
89 91
|
sseldd |
|- ( ( ph /\ x e. G ) -> ( 1st ` x ) e. X ) |
93 |
62
|
adantl |
|- ( ( ph /\ x e. G ) -> ( 2nd ` x ) e. NN0 ) |
94 |
|
oveq1 |
|- ( z = ( 1st ` x ) -> ( z ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) = ( ( 1st ` x ) ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) ) |
95 |
|
oveq2 |
|- ( m = ( 2nd ` x ) -> ( 2 ^ m ) = ( 2 ^ ( 2nd ` x ) ) ) |
96 |
95
|
oveq2d |
|- ( m = ( 2nd ` x ) -> ( 1 / ( 2 ^ m ) ) = ( 1 / ( 2 ^ ( 2nd ` x ) ) ) ) |
97 |
96
|
oveq2d |
|- ( m = ( 2nd ` x ) -> ( ( 1st ` x ) ( ball ` D ) ( 1 / ( 2 ^ m ) ) ) = ( ( 1st ` x ) ( ball ` D ) ( 1 / ( 2 ^ ( 2nd ` x ) ) ) ) ) |
98 |
|
ovex |
|- ( ( 1st ` x ) ( ball ` D ) ( 1 / ( 2 ^ ( 2nd ` x ) ) ) ) e. _V |
99 |
94 97 4 98
|
ovmpo |
|- ( ( ( 1st ` x ) e. X /\ ( 2nd ` x ) e. NN0 ) -> ( ( 1st ` x ) B ( 2nd ` x ) ) = ( ( 1st ` x ) ( ball ` D ) ( 1 / ( 2 ^ ( 2nd ` x ) ) ) ) ) |
100 |
92 93 99
|
syl2anc |
|- ( ( ph /\ x e. G ) -> ( ( 1st ` x ) B ( 2nd ` x ) ) = ( ( 1st ` x ) ( ball ` D ) ( 1 / ( 2 ^ ( 2nd ` x ) ) ) ) ) |
101 |
84 100
|
eqtrd |
|- ( ( ph /\ x e. G ) -> ( B ` x ) = ( ( 1st ` x ) ( ball ` D ) ( 1 / ( 2 ^ ( 2nd ` x ) ) ) ) ) |
102 |
|
cmetmet |
|- ( D e. ( CMet ` X ) -> D e. ( Met ` X ) ) |
103 |
5 102
|
syl |
|- ( ph -> D e. ( Met ` X ) ) |
104 |
|
metxmet |
|- ( D e. ( Met ` X ) -> D e. ( *Met ` X ) ) |
105 |
103 104
|
syl |
|- ( ph -> D e. ( *Met ` X ) ) |
106 |
105
|
adantr |
|- ( ( ph /\ x e. G ) -> D e. ( *Met ` X ) ) |
107 |
|
2nn |
|- 2 e. NN |
108 |
|
nnexpcl |
|- ( ( 2 e. NN /\ ( 2nd ` x ) e. NN0 ) -> ( 2 ^ ( 2nd ` x ) ) e. NN ) |
109 |
107 93 108
|
sylancr |
|- ( ( ph /\ x e. G ) -> ( 2 ^ ( 2nd ` x ) ) e. NN ) |
110 |
109
|
nnrpd |
|- ( ( ph /\ x e. G ) -> ( 2 ^ ( 2nd ` x ) ) e. RR+ ) |
111 |
110
|
rpreccld |
|- ( ( ph /\ x e. G ) -> ( 1 / ( 2 ^ ( 2nd ` x ) ) ) e. RR+ ) |
112 |
111
|
rpxrd |
|- ( ( ph /\ x e. G ) -> ( 1 / ( 2 ^ ( 2nd ` x ) ) ) e. RR* ) |
113 |
|
blssm |
|- ( ( D e. ( *Met ` X ) /\ ( 1st ` x ) e. X /\ ( 1 / ( 2 ^ ( 2nd ` x ) ) ) e. RR* ) -> ( ( 1st ` x ) ( ball ` D ) ( 1 / ( 2 ^ ( 2nd ` x ) ) ) ) C_ X ) |
114 |
106 92 112 113
|
syl3anc |
|- ( ( ph /\ x e. G ) -> ( ( 1st ` x ) ( ball ` D ) ( 1 / ( 2 ^ ( 2nd ` x ) ) ) ) C_ X ) |
115 |
101 114
|
eqsstrd |
|- ( ( ph /\ x e. G ) -> ( B ` x ) C_ X ) |
116 |
|
df-ss |
|- ( ( B ` x ) C_ X <-> ( ( B ` x ) i^i X ) = ( B ` x ) ) |
117 |
115 116
|
sylib |
|- ( ( ph /\ x e. G ) -> ( ( B ` x ) i^i X ) = ( B ` x ) ) |
118 |
80 117
|
eqtr3d |
|- ( ( ph /\ x e. G ) -> ( ( B ` x ) i^i U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( t B ( ( 2nd ` x ) + 1 ) ) ) = ( B ` x ) ) |
119 |
68 118
|
syl5eq |
|- ( ( ph /\ x e. G ) -> U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) = ( B ` x ) ) |
120 |
|
eqimss2 |
|- ( U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) = ( B ` x ) -> ( B ` x ) C_ U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) ) |
121 |
119 120
|
syl |
|- ( ( ph /\ x e. G ) -> ( B ` x ) C_ U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) ) |
122 |
23
|
simp3d |
|- ( x e. G -> ( ( 1st ` x ) B ( 2nd ` x ) ) e. K ) |
123 |
83 122
|
eqeltrd |
|- ( x e. G -> ( B ` x ) e. K ) |
124 |
123
|
adantl |
|- ( ( ph /\ x e. G ) -> ( B ` x ) e. K ) |
125 |
|
fvex |
|- ( B ` x ) e. _V |
126 |
125
|
inex1 |
|- ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. _V |
127 |
1 2 126
|
heiborlem1 |
|- ( ( ( F ` ( ( 2nd ` x ) + 1 ) ) e. Fin /\ ( B ` x ) C_ U_ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) /\ ( B ` x ) e. K ) -> E. t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) |
128 |
67 121 124 127
|
syl3anc |
|- ( ( ph /\ x e. G ) -> E. t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) |
129 |
85 66
|
sseldi |
|- ( ( ph /\ x e. G ) -> ( F ` ( ( 2nd ` x ) + 1 ) ) e. ~P X ) |
130 |
129
|
elpwid |
|- ( ( ph /\ x e. G ) -> ( F ` ( ( 2nd ` x ) + 1 ) ) C_ X ) |
131 |
1
|
mopnuni |
|- ( D e. ( *Met ` X ) -> X = U. J ) |
132 |
105 131
|
syl |
|- ( ph -> X = U. J ) |
133 |
132
|
adantr |
|- ( ( ph /\ x e. G ) -> X = U. J ) |
134 |
130 133
|
sseqtrd |
|- ( ( ph /\ x e. G ) -> ( F ` ( ( 2nd ` x ) + 1 ) ) C_ U. J ) |
135 |
134
|
sselda |
|- ( ( ( ph /\ x e. G ) /\ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ) -> t e. U. J ) |
136 |
135
|
adantrr |
|- ( ( ( ph /\ x e. G ) /\ ( t e. ( F ` ( ( 2nd ` x ) + 1 ) ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) -> t e. U. J ) |
137 |
64
|
adantl |
|- ( ( ph /\ x e. G ) -> ( ( 2nd ` x ) + 1 ) e. NN0 ) |
138 |
|
id |
|- ( t e. ( F ` ( ( 2nd ` x ) + 1 ) ) -> t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ) |
139 |
|
snfi |
|- { ( t B ( ( 2nd ` x ) + 1 ) ) } e. Fin |
140 |
|
inss2 |
|- ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) C_ ( t B ( ( 2nd ` x ) + 1 ) ) |
141 |
|
ovex |
|- ( t B ( ( 2nd ` x ) + 1 ) ) e. _V |
142 |
141
|
unisn |
|- U. { ( t B ( ( 2nd ` x ) + 1 ) ) } = ( t B ( ( 2nd ` x ) + 1 ) ) |
143 |
|
uniiun |
|- U. { ( t B ( ( 2nd ` x ) + 1 ) ) } = U_ g e. { ( t B ( ( 2nd ` x ) + 1 ) ) } g |
144 |
142 143
|
eqtr3i |
|- ( t B ( ( 2nd ` x ) + 1 ) ) = U_ g e. { ( t B ( ( 2nd ` x ) + 1 ) ) } g |
145 |
140 144
|
sseqtri |
|- ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) C_ U_ g e. { ( t B ( ( 2nd ` x ) + 1 ) ) } g |
146 |
|
vex |
|- g e. _V |
147 |
1 2 146
|
heiborlem1 |
|- ( ( { ( t B ( ( 2nd ` x ) + 1 ) ) } e. Fin /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) C_ U_ g e. { ( t B ( ( 2nd ` x ) + 1 ) ) } g /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) -> E. g e. { ( t B ( ( 2nd ` x ) + 1 ) ) } g e. K ) |
148 |
139 145 147
|
mp3an12 |
|- ( ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K -> E. g e. { ( t B ( ( 2nd ` x ) + 1 ) ) } g e. K ) |
149 |
|
eleq1 |
|- ( g = ( t B ( ( 2nd ` x ) + 1 ) ) -> ( g e. K <-> ( t B ( ( 2nd ` x ) + 1 ) ) e. K ) ) |
150 |
141 149
|
rexsn |
|- ( E. g e. { ( t B ( ( 2nd ` x ) + 1 ) ) } g e. K <-> ( t B ( ( 2nd ` x ) + 1 ) ) e. K ) |
151 |
148 150
|
sylib |
|- ( ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K -> ( t B ( ( 2nd ` x ) + 1 ) ) e. K ) |
152 |
|
ovex |
|- ( ( 2nd ` x ) + 1 ) e. _V |
153 |
1 2 3 46 152
|
heiborlem2 |
|- ( t G ( ( 2nd ` x ) + 1 ) <-> ( ( ( 2nd ` x ) + 1 ) e. NN0 /\ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) /\ ( t B ( ( 2nd ` x ) + 1 ) ) e. K ) ) |
154 |
153
|
biimpri |
|- ( ( ( ( 2nd ` x ) + 1 ) e. NN0 /\ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) /\ ( t B ( ( 2nd ` x ) + 1 ) ) e. K ) -> t G ( ( 2nd ` x ) + 1 ) ) |
155 |
137 138 151 154
|
syl3an |
|- ( ( ( ph /\ x e. G ) /\ t e. ( F ` ( ( 2nd ` x ) + 1 ) ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) -> t G ( ( 2nd ` x ) + 1 ) ) |
156 |
155
|
3expb |
|- ( ( ( ph /\ x e. G ) /\ ( t e. ( F ` ( ( 2nd ` x ) + 1 ) ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) -> t G ( ( 2nd ` x ) + 1 ) ) |
157 |
|
simprr |
|- ( ( ( ph /\ x e. G ) /\ ( t e. ( F ` ( ( 2nd ` x ) + 1 ) ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) -> ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) |
158 |
136 156 157
|
jca32 |
|- ( ( ( ph /\ x e. G ) /\ ( t e. ( F ` ( ( 2nd ` x ) + 1 ) ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) -> ( t e. U. J /\ ( t G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) ) |
159 |
158
|
ex |
|- ( ( ph /\ x e. G ) -> ( ( t e. ( F ` ( ( 2nd ` x ) + 1 ) ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) -> ( t e. U. J /\ ( t G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) ) ) |
160 |
159
|
reximdv2 |
|- ( ( ph /\ x e. G ) -> ( E. t e. ( F ` ( ( 2nd ` x ) + 1 ) ) ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K -> E. t e. U. J ( t G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) ) |
161 |
128 160
|
mpd |
|- ( ( ph /\ x e. G ) -> E. t e. U. J ( t G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) |
162 |
161
|
ralrimiva |
|- ( ph -> A. x e. G E. t e. U. J ( t G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) |
163 |
1
|
fvexi |
|- J e. _V |
164 |
163
|
uniex |
|- U. J e. _V |
165 |
|
breq1 |
|- ( t = ( g ` x ) -> ( t G ( ( 2nd ` x ) + 1 ) <-> ( g ` x ) G ( ( 2nd ` x ) + 1 ) ) ) |
166 |
|
oveq1 |
|- ( t = ( g ` x ) -> ( t B ( ( 2nd ` x ) + 1 ) ) = ( ( g ` x ) B ( ( 2nd ` x ) + 1 ) ) ) |
167 |
166
|
ineq2d |
|- ( t = ( g ` x ) -> ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) = ( ( B ` x ) i^i ( ( g ` x ) B ( ( 2nd ` x ) + 1 ) ) ) ) |
168 |
167
|
eleq1d |
|- ( t = ( g ` x ) -> ( ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K <-> ( ( B ` x ) i^i ( ( g ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) |
169 |
165 168
|
anbi12d |
|- ( t = ( g ` x ) -> ( ( t G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) <-> ( ( g ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( g ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) ) |
170 |
164 169
|
axcc4dom |
|- ( ( G ~<_ _om /\ A. x e. G E. t e. U. J ( t G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( t B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) -> E. g ( g : G --> U. J /\ A. x e. G ( ( g ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( g ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) ) |
171 |
61 162 170
|
syl2anc |
|- ( ph -> E. g ( g : G --> U. J /\ A. x e. G ( ( g ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( g ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) ) |
172 |
|
exsimpr |
|- ( E. g ( g : G --> U. J /\ A. x e. G ( ( g ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( g ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) -> E. g A. x e. G ( ( g ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( g ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) |
173 |
171 172
|
syl |
|- ( ph -> E. g A. x e. G ( ( g ` x ) G ( ( 2nd ` x ) + 1 ) /\ ( ( B ` x ) i^i ( ( g ` x ) B ( ( 2nd ` x ) + 1 ) ) ) e. K ) ) |