Step |
Hyp |
Ref |
Expression |
1 |
|
bndth.1 |
|- X = U. J |
2 |
|
bndth.2 |
|- K = ( topGen ` ran (,) ) |
3 |
|
bndth.3 |
|- ( ph -> J e. Comp ) |
4 |
|
bndth.4 |
|- ( ph -> F e. ( J Cn K ) ) |
5 |
|
retopon |
|- ( topGen ` ran (,) ) e. ( TopOn ` RR ) |
6 |
2 5
|
eqeltri |
|- K e. ( TopOn ` RR ) |
7 |
6
|
toponunii |
|- RR = U. K |
8 |
1 7
|
cnf |
|- ( F e. ( J Cn K ) -> F : X --> RR ) |
9 |
4 8
|
syl |
|- ( ph -> F : X --> RR ) |
10 |
9
|
frnd |
|- ( ph -> ran F C_ RR ) |
11 |
|
unieq |
|- ( u = ( (,) " ( { -oo } X. RR ) ) -> U. u = U. ( (,) " ( { -oo } X. RR ) ) ) |
12 |
|
imassrn |
|- ( (,) " ( { -oo } X. RR ) ) C_ ran (,) |
13 |
12
|
unissi |
|- U. ( (,) " ( { -oo } X. RR ) ) C_ U. ran (,) |
14 |
|
unirnioo |
|- RR = U. ran (,) |
15 |
13 14
|
sseqtrri |
|- U. ( (,) " ( { -oo } X. RR ) ) C_ RR |
16 |
|
id |
|- ( x e. RR -> x e. RR ) |
17 |
|
ltp1 |
|- ( x e. RR -> x < ( x + 1 ) ) |
18 |
|
ressxr |
|- RR C_ RR* |
19 |
|
peano2re |
|- ( x e. RR -> ( x + 1 ) e. RR ) |
20 |
18 19
|
sselid |
|- ( x e. RR -> ( x + 1 ) e. RR* ) |
21 |
|
elioomnf |
|- ( ( x + 1 ) e. RR* -> ( x e. ( -oo (,) ( x + 1 ) ) <-> ( x e. RR /\ x < ( x + 1 ) ) ) ) |
22 |
20 21
|
syl |
|- ( x e. RR -> ( x e. ( -oo (,) ( x + 1 ) ) <-> ( x e. RR /\ x < ( x + 1 ) ) ) ) |
23 |
16 17 22
|
mpbir2and |
|- ( x e. RR -> x e. ( -oo (,) ( x + 1 ) ) ) |
24 |
|
df-ov |
|- ( -oo (,) ( x + 1 ) ) = ( (,) ` <. -oo , ( x + 1 ) >. ) |
25 |
|
mnfxr |
|- -oo e. RR* |
26 |
25
|
elexi |
|- -oo e. _V |
27 |
26
|
snid |
|- -oo e. { -oo } |
28 |
|
opelxpi |
|- ( ( -oo e. { -oo } /\ ( x + 1 ) e. RR ) -> <. -oo , ( x + 1 ) >. e. ( { -oo } X. RR ) ) |
29 |
27 19 28
|
sylancr |
|- ( x e. RR -> <. -oo , ( x + 1 ) >. e. ( { -oo } X. RR ) ) |
30 |
|
ioof |
|- (,) : ( RR* X. RR* ) --> ~P RR |
31 |
|
ffun |
|- ( (,) : ( RR* X. RR* ) --> ~P RR -> Fun (,) ) |
32 |
30 31
|
ax-mp |
|- Fun (,) |
33 |
|
snssi |
|- ( -oo e. RR* -> { -oo } C_ RR* ) |
34 |
25 33
|
ax-mp |
|- { -oo } C_ RR* |
35 |
|
xpss12 |
|- ( ( { -oo } C_ RR* /\ RR C_ RR* ) -> ( { -oo } X. RR ) C_ ( RR* X. RR* ) ) |
36 |
34 18 35
|
mp2an |
|- ( { -oo } X. RR ) C_ ( RR* X. RR* ) |
37 |
30
|
fdmi |
|- dom (,) = ( RR* X. RR* ) |
38 |
36 37
|
sseqtrri |
|- ( { -oo } X. RR ) C_ dom (,) |
39 |
|
funfvima2 |
|- ( ( Fun (,) /\ ( { -oo } X. RR ) C_ dom (,) ) -> ( <. -oo , ( x + 1 ) >. e. ( { -oo } X. RR ) -> ( (,) ` <. -oo , ( x + 1 ) >. ) e. ( (,) " ( { -oo } X. RR ) ) ) ) |
40 |
32 38 39
|
mp2an |
|- ( <. -oo , ( x + 1 ) >. e. ( { -oo } X. RR ) -> ( (,) ` <. -oo , ( x + 1 ) >. ) e. ( (,) " ( { -oo } X. RR ) ) ) |
41 |
29 40
|
syl |
|- ( x e. RR -> ( (,) ` <. -oo , ( x + 1 ) >. ) e. ( (,) " ( { -oo } X. RR ) ) ) |
42 |
24 41
|
eqeltrid |
|- ( x e. RR -> ( -oo (,) ( x + 1 ) ) e. ( (,) " ( { -oo } X. RR ) ) ) |
43 |
|
elunii |
|- ( ( x e. ( -oo (,) ( x + 1 ) ) /\ ( -oo (,) ( x + 1 ) ) e. ( (,) " ( { -oo } X. RR ) ) ) -> x e. U. ( (,) " ( { -oo } X. RR ) ) ) |
44 |
23 42 43
|
syl2anc |
|- ( x e. RR -> x e. U. ( (,) " ( { -oo } X. RR ) ) ) |
45 |
44
|
ssriv |
|- RR C_ U. ( (,) " ( { -oo } X. RR ) ) |
46 |
15 45
|
eqssi |
|- U. ( (,) " ( { -oo } X. RR ) ) = RR |
47 |
11 46
|
eqtrdi |
|- ( u = ( (,) " ( { -oo } X. RR ) ) -> U. u = RR ) |
48 |
47
|
sseq2d |
|- ( u = ( (,) " ( { -oo } X. RR ) ) -> ( ran F C_ U. u <-> ran F C_ RR ) ) |
49 |
|
pweq |
|- ( u = ( (,) " ( { -oo } X. RR ) ) -> ~P u = ~P ( (,) " ( { -oo } X. RR ) ) ) |
50 |
49
|
ineq1d |
|- ( u = ( (,) " ( { -oo } X. RR ) ) -> ( ~P u i^i Fin ) = ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) |
51 |
50
|
rexeqdv |
|- ( u = ( (,) " ( { -oo } X. RR ) ) -> ( E. v e. ( ~P u i^i Fin ) ran F C_ U. v <-> E. v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ran F C_ U. v ) ) |
52 |
48 51
|
imbi12d |
|- ( u = ( (,) " ( { -oo } X. RR ) ) -> ( ( ran F C_ U. u -> E. v e. ( ~P u i^i Fin ) ran F C_ U. v ) <-> ( ran F C_ RR -> E. v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ran F C_ U. v ) ) ) |
53 |
|
rncmp |
|- ( ( J e. Comp /\ F e. ( J Cn K ) ) -> ( K |`t ran F ) e. Comp ) |
54 |
3 4 53
|
syl2anc |
|- ( ph -> ( K |`t ran F ) e. Comp ) |
55 |
|
retop |
|- ( topGen ` ran (,) ) e. Top |
56 |
2 55
|
eqeltri |
|- K e. Top |
57 |
7
|
cmpsub |
|- ( ( K e. Top /\ ran F C_ RR ) -> ( ( K |`t ran F ) e. Comp <-> A. u e. ~P K ( ran F C_ U. u -> E. v e. ( ~P u i^i Fin ) ran F C_ U. v ) ) ) |
58 |
56 10 57
|
sylancr |
|- ( ph -> ( ( K |`t ran F ) e. Comp <-> A. u e. ~P K ( ran F C_ U. u -> E. v e. ( ~P u i^i Fin ) ran F C_ U. v ) ) ) |
59 |
54 58
|
mpbid |
|- ( ph -> A. u e. ~P K ( ran F C_ U. u -> E. v e. ( ~P u i^i Fin ) ran F C_ U. v ) ) |
60 |
|
retopbas |
|- ran (,) e. TopBases |
61 |
|
bastg |
|- ( ran (,) e. TopBases -> ran (,) C_ ( topGen ` ran (,) ) ) |
62 |
60 61
|
ax-mp |
|- ran (,) C_ ( topGen ` ran (,) ) |
63 |
62 2
|
sseqtrri |
|- ran (,) C_ K |
64 |
12 63
|
sstri |
|- ( (,) " ( { -oo } X. RR ) ) C_ K |
65 |
56 64
|
elpwi2 |
|- ( (,) " ( { -oo } X. RR ) ) e. ~P K |
66 |
65
|
a1i |
|- ( ph -> ( (,) " ( { -oo } X. RR ) ) e. ~P K ) |
67 |
52 59 66
|
rspcdva |
|- ( ph -> ( ran F C_ RR -> E. v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ran F C_ U. v ) ) |
68 |
10 67
|
mpd |
|- ( ph -> E. v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ran F C_ U. v ) |
69 |
|
simpr |
|- ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) -> v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) |
70 |
|
elin |
|- ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) <-> ( v e. ~P ( (,) " ( { -oo } X. RR ) ) /\ v e. Fin ) ) |
71 |
69 70
|
sylib |
|- ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) -> ( v e. ~P ( (,) " ( { -oo } X. RR ) ) /\ v e. Fin ) ) |
72 |
71
|
adantrr |
|- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> ( v e. ~P ( (,) " ( { -oo } X. RR ) ) /\ v e. Fin ) ) |
73 |
72
|
simprd |
|- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> v e. Fin ) |
74 |
71
|
simpld |
|- ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) -> v e. ~P ( (,) " ( { -oo } X. RR ) ) ) |
75 |
74
|
elpwid |
|- ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) -> v C_ ( (,) " ( { -oo } X. RR ) ) ) |
76 |
34
|
sseli |
|- ( u e. { -oo } -> u e. RR* ) |
77 |
76
|
adantr |
|- ( ( u e. { -oo } /\ w e. RR ) -> u e. RR* ) |
78 |
18
|
sseli |
|- ( w e. RR -> w e. RR* ) |
79 |
78
|
adantl |
|- ( ( u e. { -oo } /\ w e. RR ) -> w e. RR* ) |
80 |
|
mnflt |
|- ( w e. RR -> -oo < w ) |
81 |
|
xrltnle |
|- ( ( -oo e. RR* /\ w e. RR* ) -> ( -oo < w <-> -. w <_ -oo ) ) |
82 |
25 78 81
|
sylancr |
|- ( w e. RR -> ( -oo < w <-> -. w <_ -oo ) ) |
83 |
80 82
|
mpbid |
|- ( w e. RR -> -. w <_ -oo ) |
84 |
83
|
adantl |
|- ( ( u e. { -oo } /\ w e. RR ) -> -. w <_ -oo ) |
85 |
|
elsni |
|- ( u e. { -oo } -> u = -oo ) |
86 |
85
|
adantr |
|- ( ( u e. { -oo } /\ w e. RR ) -> u = -oo ) |
87 |
86
|
breq2d |
|- ( ( u e. { -oo } /\ w e. RR ) -> ( w <_ u <-> w <_ -oo ) ) |
88 |
84 87
|
mtbird |
|- ( ( u e. { -oo } /\ w e. RR ) -> -. w <_ u ) |
89 |
|
ioo0 |
|- ( ( u e. RR* /\ w e. RR* ) -> ( ( u (,) w ) = (/) <-> w <_ u ) ) |
90 |
76 78 89
|
syl2an |
|- ( ( u e. { -oo } /\ w e. RR ) -> ( ( u (,) w ) = (/) <-> w <_ u ) ) |
91 |
90
|
necon3abid |
|- ( ( u e. { -oo } /\ w e. RR ) -> ( ( u (,) w ) =/= (/) <-> -. w <_ u ) ) |
92 |
88 91
|
mpbird |
|- ( ( u e. { -oo } /\ w e. RR ) -> ( u (,) w ) =/= (/) ) |
93 |
|
df-ioo |
|- (,) = ( y e. RR* , z e. RR* |-> { v e. RR* | ( y < v /\ v < z ) } ) |
94 |
|
idd |
|- ( ( x e. RR* /\ w e. RR* ) -> ( x < w -> x < w ) ) |
95 |
|
xrltle |
|- ( ( x e. RR* /\ w e. RR* ) -> ( x < w -> x <_ w ) ) |
96 |
|
idd |
|- ( ( u e. RR* /\ x e. RR* ) -> ( u < x -> u < x ) ) |
97 |
|
xrltle |
|- ( ( u e. RR* /\ x e. RR* ) -> ( u < x -> u <_ x ) ) |
98 |
93 94 95 96 97
|
ixxub |
|- ( ( u e. RR* /\ w e. RR* /\ ( u (,) w ) =/= (/) ) -> sup ( ( u (,) w ) , RR* , < ) = w ) |
99 |
77 79 92 98
|
syl3anc |
|- ( ( u e. { -oo } /\ w e. RR ) -> sup ( ( u (,) w ) , RR* , < ) = w ) |
100 |
|
simpr |
|- ( ( u e. { -oo } /\ w e. RR ) -> w e. RR ) |
101 |
99 100
|
eqeltrd |
|- ( ( u e. { -oo } /\ w e. RR ) -> sup ( ( u (,) w ) , RR* , < ) e. RR ) |
102 |
101
|
rgen2 |
|- A. u e. { -oo } A. w e. RR sup ( ( u (,) w ) , RR* , < ) e. RR |
103 |
|
fveq2 |
|- ( z = <. u , w >. -> ( (,) ` z ) = ( (,) ` <. u , w >. ) ) |
104 |
|
df-ov |
|- ( u (,) w ) = ( (,) ` <. u , w >. ) |
105 |
103 104
|
eqtr4di |
|- ( z = <. u , w >. -> ( (,) ` z ) = ( u (,) w ) ) |
106 |
105
|
supeq1d |
|- ( z = <. u , w >. -> sup ( ( (,) ` z ) , RR* , < ) = sup ( ( u (,) w ) , RR* , < ) ) |
107 |
106
|
eleq1d |
|- ( z = <. u , w >. -> ( sup ( ( (,) ` z ) , RR* , < ) e. RR <-> sup ( ( u (,) w ) , RR* , < ) e. RR ) ) |
108 |
107
|
ralxp |
|- ( A. z e. ( { -oo } X. RR ) sup ( ( (,) ` z ) , RR* , < ) e. RR <-> A. u e. { -oo } A. w e. RR sup ( ( u (,) w ) , RR* , < ) e. RR ) |
109 |
102 108
|
mpbir |
|- A. z e. ( { -oo } X. RR ) sup ( ( (,) ` z ) , RR* , < ) e. RR |
110 |
|
ffn |
|- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) ) |
111 |
30 110
|
ax-mp |
|- (,) Fn ( RR* X. RR* ) |
112 |
|
supeq1 |
|- ( w = ( (,) ` z ) -> sup ( w , RR* , < ) = sup ( ( (,) ` z ) , RR* , < ) ) |
113 |
112
|
eleq1d |
|- ( w = ( (,) ` z ) -> ( sup ( w , RR* , < ) e. RR <-> sup ( ( (,) ` z ) , RR* , < ) e. RR ) ) |
114 |
113
|
ralima |
|- ( ( (,) Fn ( RR* X. RR* ) /\ ( { -oo } X. RR ) C_ ( RR* X. RR* ) ) -> ( A. w e. ( (,) " ( { -oo } X. RR ) ) sup ( w , RR* , < ) e. RR <-> A. z e. ( { -oo } X. RR ) sup ( ( (,) ` z ) , RR* , < ) e. RR ) ) |
115 |
111 36 114
|
mp2an |
|- ( A. w e. ( (,) " ( { -oo } X. RR ) ) sup ( w , RR* , < ) e. RR <-> A. z e. ( { -oo } X. RR ) sup ( ( (,) ` z ) , RR* , < ) e. RR ) |
116 |
109 115
|
mpbir |
|- A. w e. ( (,) " ( { -oo } X. RR ) ) sup ( w , RR* , < ) e. RR |
117 |
|
ssralv |
|- ( v C_ ( (,) " ( { -oo } X. RR ) ) -> ( A. w e. ( (,) " ( { -oo } X. RR ) ) sup ( w , RR* , < ) e. RR -> A. w e. v sup ( w , RR* , < ) e. RR ) ) |
118 |
75 116 117
|
mpisyl |
|- ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) -> A. w e. v sup ( w , RR* , < ) e. RR ) |
119 |
118
|
adantrr |
|- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> A. w e. v sup ( w , RR* , < ) e. RR ) |
120 |
|
fimaxre3 |
|- ( ( v e. Fin /\ A. w e. v sup ( w , RR* , < ) e. RR ) -> E. x e. RR A. w e. v sup ( w , RR* , < ) <_ x ) |
121 |
73 119 120
|
syl2anc |
|- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> E. x e. RR A. w e. v sup ( w , RR* , < ) <_ x ) |
122 |
|
simplrr |
|- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ran F C_ U. v ) |
123 |
122
|
sselda |
|- ( ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) /\ z e. ran F ) -> z e. U. v ) |
124 |
|
eluni2 |
|- ( z e. U. v <-> E. w e. v z e. w ) |
125 |
|
r19.29r |
|- ( ( E. w e. v z e. w /\ A. w e. v sup ( w , RR* , < ) <_ x ) -> E. w e. v ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) |
126 |
|
sspwuni |
|- ( ( (,) " ( { -oo } X. RR ) ) C_ ~P RR <-> U. ( (,) " ( { -oo } X. RR ) ) C_ RR ) |
127 |
15 126
|
mpbir |
|- ( (,) " ( { -oo } X. RR ) ) C_ ~P RR |
128 |
75
|
3ad2ant1 |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> v C_ ( (,) " ( { -oo } X. RR ) ) ) |
129 |
|
simp2r |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> w e. v ) |
130 |
128 129
|
sseldd |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> w e. ( (,) " ( { -oo } X. RR ) ) ) |
131 |
127 130
|
sselid |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> w e. ~P RR ) |
132 |
131
|
elpwid |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> w C_ RR ) |
133 |
|
simp3l |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> z e. w ) |
134 |
132 133
|
sseldd |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> z e. RR ) |
135 |
118
|
r19.21bi |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ w e. v ) -> sup ( w , RR* , < ) e. RR ) |
136 |
135
|
adantrl |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) ) -> sup ( w , RR* , < ) e. RR ) |
137 |
136
|
3adant3 |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> sup ( w , RR* , < ) e. RR ) |
138 |
|
simp2l |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> x e. RR ) |
139 |
132 18
|
sstrdi |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> w C_ RR* ) |
140 |
|
supxrub |
|- ( ( w C_ RR* /\ z e. w ) -> z <_ sup ( w , RR* , < ) ) |
141 |
139 133 140
|
syl2anc |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> z <_ sup ( w , RR* , < ) ) |
142 |
|
simp3r |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> sup ( w , RR* , < ) <_ x ) |
143 |
134 137 138 141 142
|
letrd |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) /\ ( z e. w /\ sup ( w , RR* , < ) <_ x ) ) -> z <_ x ) |
144 |
143
|
3expia |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ ( x e. RR /\ w e. v ) ) -> ( ( z e. w /\ sup ( w , RR* , < ) <_ x ) -> z <_ x ) ) |
145 |
144
|
anassrs |
|- ( ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ x e. RR ) /\ w e. v ) -> ( ( z e. w /\ sup ( w , RR* , < ) <_ x ) -> z <_ x ) ) |
146 |
145
|
rexlimdva |
|- ( ( ( ph /\ v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) ) /\ x e. RR ) -> ( E. w e. v ( z e. w /\ sup ( w , RR* , < ) <_ x ) -> z <_ x ) ) |
147 |
146
|
adantlrr |
|- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ( E. w e. v ( z e. w /\ sup ( w , RR* , < ) <_ x ) -> z <_ x ) ) |
148 |
125 147
|
syl5 |
|- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ( ( E. w e. v z e. w /\ A. w e. v sup ( w , RR* , < ) <_ x ) -> z <_ x ) ) |
149 |
148
|
expdimp |
|- ( ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) /\ E. w e. v z e. w ) -> ( A. w e. v sup ( w , RR* , < ) <_ x -> z <_ x ) ) |
150 |
124 149
|
sylan2b |
|- ( ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) /\ z e. U. v ) -> ( A. w e. v sup ( w , RR* , < ) <_ x -> z <_ x ) ) |
151 |
123 150
|
syldan |
|- ( ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) /\ z e. ran F ) -> ( A. w e. v sup ( w , RR* , < ) <_ x -> z <_ x ) ) |
152 |
151
|
ralrimdva |
|- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ( A. w e. v sup ( w , RR* , < ) <_ x -> A. z e. ran F z <_ x ) ) |
153 |
9
|
ffnd |
|- ( ph -> F Fn X ) |
154 |
153
|
ad2antrr |
|- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> F Fn X ) |
155 |
|
breq1 |
|- ( z = ( F ` y ) -> ( z <_ x <-> ( F ` y ) <_ x ) ) |
156 |
155
|
ralrn |
|- ( F Fn X -> ( A. z e. ran F z <_ x <-> A. y e. X ( F ` y ) <_ x ) ) |
157 |
154 156
|
syl |
|- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ( A. z e. ran F z <_ x <-> A. y e. X ( F ` y ) <_ x ) ) |
158 |
152 157
|
sylibd |
|- ( ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) /\ x e. RR ) -> ( A. w e. v sup ( w , RR* , < ) <_ x -> A. y e. X ( F ` y ) <_ x ) ) |
159 |
158
|
reximdva |
|- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> ( E. x e. RR A. w e. v sup ( w , RR* , < ) <_ x -> E. x e. RR A. y e. X ( F ` y ) <_ x ) ) |
160 |
121 159
|
mpd |
|- ( ( ph /\ ( v e. ( ~P ( (,) " ( { -oo } X. RR ) ) i^i Fin ) /\ ran F C_ U. v ) ) -> E. x e. RR A. y e. X ( F ` y ) <_ x ) |
161 |
68 160
|
rexlimddv |
|- ( ph -> E. x e. RR A. y e. X ( F ` y ) <_ x ) |