Step |
Hyp |
Ref |
Expression |
1 |
|
choicefi.a |
|
2 |
|
choicefi.b |
|
3 |
|
choicefi.n |
|
4 |
|
mptfi |
|
5 |
|
rnfi |
|
6 |
|
fnchoice |
|
7 |
1 4 5 6
|
4syl |
|
8 |
|
simpl |
|
9 |
|
simprl |
|
10 |
|
nfv |
|
11 |
|
nfra1 |
|
12 |
10 11
|
nfan |
|
13 |
|
rspa |
|
14 |
13
|
adantll |
|
15 |
|
vex |
|
16 |
|
eqid |
|
17 |
16
|
elrnmpt |
|
18 |
15 17
|
ax-mp |
|
19 |
18
|
biimpi |
|
20 |
19
|
adantl |
|
21 |
|
simp3 |
|
22 |
3
|
3adant3 |
|
23 |
21 22
|
eqnetrd |
|
24 |
23
|
3exp |
|
25 |
24
|
rexlimdv |
|
26 |
25
|
adantr |
|
27 |
20 26
|
mpd |
|
28 |
27
|
adantlr |
|
29 |
|
id |
|
30 |
29
|
imp |
|
31 |
14 28 30
|
syl2anc |
|
32 |
31
|
ex |
|
33 |
12 32
|
ralrimi |
|
34 |
|
rsp |
|
35 |
33 34
|
syl |
|
36 |
12 35
|
ralrimi |
|
37 |
36
|
adantrl |
|
38 |
|
vex |
|
39 |
38
|
a1i |
|
40 |
1
|
mptexd |
|
41 |
|
coexg |
|
42 |
39 40 41
|
syl2anc |
|
43 |
42
|
3ad2ant1 |
|
44 |
|
simpr |
|
45 |
2
|
ralrimiva |
|
46 |
16
|
fnmpt |
|
47 |
45 46
|
syl |
|
48 |
47
|
adantr |
|
49 |
|
ssidd |
|
50 |
|
fnco |
|
51 |
44 48 49 50
|
syl3anc |
|
52 |
51
|
3adant3 |
|
53 |
|
nfv |
|
54 |
|
nfcv |
|
55 |
|
nfmpt1 |
|
56 |
55
|
nfrn |
|
57 |
54 56
|
nffn |
|
58 |
|
nfv |
|
59 |
56 58
|
nfralw |
|
60 |
53 57 59
|
nf3an |
|
61 |
|
funmpt |
|
62 |
61
|
a1i |
|
63 |
|
simpr |
|
64 |
16 2
|
dmmptd |
|
65 |
64
|
eqcomd |
|
66 |
65
|
adantr |
|
67 |
63 66
|
eleqtrd |
|
68 |
|
fvco |
|
69 |
62 67 68
|
syl2anc |
|
70 |
16
|
fvmpt2 |
|
71 |
63 2 70
|
syl2anc |
|
72 |
71
|
fveq2d |
|
73 |
69 72
|
eqtrd |
|
74 |
73
|
3ad2antl1 |
|
75 |
16
|
elrnmpt1 |
|
76 |
63 2 75
|
syl2anc |
|
77 |
76
|
3ad2antl1 |
|
78 |
|
simpl3 |
|
79 |
|
fveq2 |
|
80 |
|
id |
|
81 |
79 80
|
eleq12d |
|
82 |
81
|
rspcva |
|
83 |
77 78 82
|
syl2anc |
|
84 |
74 83
|
eqeltrd |
|
85 |
84
|
ex |
|
86 |
60 85
|
ralrimi |
|
87 |
52 86
|
jca |
|
88 |
|
fneq1 |
|
89 |
|
nfcv |
|
90 |
54 55
|
nfco |
|
91 |
89 90
|
nfeq |
|
92 |
|
fveq1 |
|
93 |
92
|
eleq1d |
|
94 |
91 93
|
ralbid |
|
95 |
88 94
|
anbi12d |
|
96 |
95
|
spcegv |
|
97 |
43 87 96
|
sylc |
|
98 |
8 9 37 97
|
syl3anc |
|
99 |
98
|
ex |
|
100 |
99
|
exlimdv |
|
101 |
7 100
|
mpd |
|