Step |
Hyp |
Ref |
Expression |
1 |
|
opnvonmbllem2.x |
|
2 |
|
opnvonmbllem2.n |
|
3 |
|
opnvonmbllem2.g |
|
4 |
|
opnvonmbl.k |
|
5 |
|
eqid |
|
6 |
5
|
rrxmetfi |
|
7 |
1 6
|
syl |
|
8 |
|
metxmet |
|
9 |
7 8
|
syl |
|
10 |
9
|
adantr |
|
11 |
|
eqid |
|
12 |
11
|
rrxval |
|
13 |
1 12
|
syl |
|
14 |
13
|
fveq2d |
|
15 |
|
ovex |
|
16 |
|
eqid |
|
17 |
|
eqid |
|
18 |
|
eqid |
|
19 |
16 17 18
|
tcphtopn |
|
20 |
15 19
|
ax-mp |
|
21 |
20
|
a1i |
|
22 |
13
|
eqcomd |
|
23 |
22
|
fveq2d |
|
24 |
23
|
fveq2d |
|
25 |
14 21 24
|
3eqtrd |
|
26 |
3 25
|
eleqtrd |
|
27 |
26
|
adantr |
|
28 |
|
simpr |
|
29 |
|
eqid |
|
30 |
29
|
mopni2 |
|
31 |
10 27 28 30
|
syl3anc |
|
32 |
1
|
ad2antrr |
|
33 |
|
eqid |
|
34 |
33
|
rrxtoponfi |
|
35 |
1 34
|
syl |
|
36 |
|
toponss |
|
37 |
35 3 36
|
syl2anc |
|
38 |
37
|
adantr |
|
39 |
38 28
|
sseldd |
|
40 |
39
|
adantr |
|
41 |
|
simpr |
|
42 |
32 40 41
|
hoiqssbl |
|
43 |
42
|
3adant3 |
|
44 |
|
nfv |
|
45 |
|
nfv |
|
46 |
|
nfcv |
|
47 |
|
nfixp1 |
|
48 |
46 47
|
nfel |
|
49 |
|
nfcv |
|
50 |
47 49
|
nfss |
|
51 |
48 50
|
nfan |
|
52 |
44 45 51
|
nf3an |
|
53 |
1
|
adantr |
|
54 |
53
|
3ad2ant1 |
|
55 |
|
elmapi |
|
56 |
55
|
adantr |
|
57 |
56
|
3ad2ant2 |
|
58 |
|
elmapi |
|
59 |
58
|
adantl |
|
60 |
59
|
3ad2ant2 |
|
61 |
|
simp3r |
|
62 |
|
simp1r |
|
63 |
|
simp3l |
|
64 |
|
eqid |
|
65 |
52 54 57 60 61 62 63 4 64
|
opnvonmbllem1 |
|
66 |
65
|
3exp |
|
67 |
66
|
adantlr |
|
68 |
67
|
3adant2 |
|
69 |
68
|
rexlimdvv |
|
70 |
43 69
|
mpd |
|
71 |
70
|
3exp |
|
72 |
71
|
rexlimdv |
|
73 |
31 72
|
mpd |
|
74 |
|
eliun |
|
75 |
73 74
|
sylibr |
|
76 |
75
|
ralrimiva |
|
77 |
|
dfss3 |
|
78 |
76 77
|
sylibr |
|
79 |
4
|
eleq2i |
|
80 |
79
|
biimpi |
|
81 |
80
|
adantl |
|
82 |
|
rabid |
|
83 |
81 82
|
sylib |
|
84 |
83
|
simprd |
|
85 |
84
|
ralrimiva |
|
86 |
|
iunss |
|
87 |
85 86
|
sylibr |
|
88 |
78 87
|
eqssd |
|
89 |
1 2
|
dmovnsal |
|
90 |
|
ssrab2 |
|
91 |
4 90
|
eqsstri |
|
92 |
91
|
a1i |
|
93 |
|
qct |
|
94 |
93
|
a1i |
|
95 |
|
xpct |
|
96 |
94 94 95
|
syl2anc |
|
97 |
96 1
|
mpct |
|
98 |
|
ssct |
|
99 |
92 97 98
|
syl2anc |
|
100 |
|
reex |
|
101 |
100 100
|
xpex |
|
102 |
|
qssre |
|
103 |
|
xpss12 |
|
104 |
102 102 103
|
mp2an |
|
105 |
|
mapss |
|
106 |
101 104 105
|
mp2an |
|
107 |
91
|
sseli |
|
108 |
106 107
|
sseldi |
|
109 |
|
elmapi |
|
110 |
108 109
|
syl |
|
111 |
110
|
adantl |
|
112 |
|
2fveq3 |
|
113 |
112
|
cbvmptv |
|
114 |
|
2fveq3 |
|
115 |
114
|
cbvmptv |
|
116 |
111 113 115
|
hoicoto2 |
|
117 |
1
|
adantr |
|
118 |
111
|
ffvelrnda |
|
119 |
|
xp1st |
|
120 |
118 119
|
syl |
|
121 |
120
|
fmpttd |
|
122 |
|
xp2nd |
|
123 |
118 122
|
syl |
|
124 |
123
|
fmpttd |
|
125 |
117 2 121 124
|
hoimbl |
|
126 |
116 125
|
eqeltrd |
|
127 |
89 99 126
|
saliuncl |
|
128 |
88 127
|
eqeltrd |
|