Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|
2 |
1
|
elrnmpt |
|
3 |
2
|
elv |
|
4 |
|
simpr |
|
5 |
|
simpl3 |
|
6 |
|
simpr |
|
7 |
5 6
|
sseldd |
|
8 |
|
inisegn0 |
|
9 |
7 8
|
sylib |
|
10 |
9
|
adantr |
|
11 |
4 10
|
eqnetrd |
|
12 |
11
|
r19.29an |
|
13 |
3 12
|
sylan2b |
|
14 |
13
|
ralrimiva |
|
15 |
|
simp2 |
|
16 |
|
simp1 |
|
17 |
15 16
|
jca |
|
18 |
|
fnex |
|
19 |
|
rnexg |
|
20 |
17 18 19
|
3syl |
|
21 |
|
simp3 |
|
22 |
20 21
|
ssexd |
|
23 |
|
mptexg |
|
24 |
|
rnexg |
|
25 |
22 23 24
|
3syl |
|
26 |
|
fvi |
|
27 |
25 26
|
syl |
|
28 |
27
|
raleqdv |
|
29 |
14 28
|
mpbird |
|
30 |
|
fvex |
|
31 |
30
|
ac5b |
|
32 |
29 31
|
syl |
|
33 |
27
|
unieqd |
|
34 |
27 33
|
feq23d |
|
35 |
27
|
raleqdv |
|
36 |
34 35
|
anbi12d |
|
37 |
36
|
exbidv |
|
38 |
32 37
|
mpbid |
|
39 |
|
vex |
|
40 |
39
|
rnex |
|
41 |
40
|
a1i |
|
42 |
|
simplr |
|
43 |
|
frn |
|
44 |
42 43
|
syl |
|
45 |
|
nfv |
|
46 |
|
nfcv |
|
47 |
|
nfmpt1 |
|
48 |
47
|
nfrn |
|
49 |
48
|
nfuni |
|
50 |
46 48 49
|
nff |
|
51 |
45 50
|
nfan |
|
52 |
|
nfv |
|
53 |
48 52
|
nfralw |
|
54 |
51 53
|
nfan |
|
55 |
17 18
|
syl |
|
56 |
55
|
ad3antrrr |
|
57 |
|
cnvexg |
|
58 |
|
imaexg |
|
59 |
56 57 58
|
3syl |
|
60 |
|
cnvimass |
|
61 |
60
|
a1i |
|
62 |
15
|
fndmd |
|
63 |
62
|
ad3antrrr |
|
64 |
61 63
|
sseqtrd |
|
65 |
59 64
|
elpwd |
|
66 |
65
|
ex |
|
67 |
54 66
|
ralrimi |
|
68 |
1
|
rnmptss |
|
69 |
67 68
|
syl |
|
70 |
|
sspwuni |
|
71 |
69 70
|
sylib |
|
72 |
44 71
|
sstrd |
|
73 |
41 72
|
elpwd |
|
74 |
|
fnfun |
|
75 |
15 74
|
syl |
|
76 |
75
|
ad5antr |
|
77 |
|
sndisj |
|
78 |
|
disjpreima |
|
79 |
76 77 78
|
sylancl |
|
80 |
|
disjrnmpt |
|
81 |
79 80
|
syl |
|
82 |
|
simpllr |
|
83 |
|
simplr |
|
84 |
|
simp-4r |
|
85 |
|
fveq2 |
|
86 |
|
id |
|
87 |
85 86
|
eleq12d |
|
88 |
87
|
rspcv |
|
89 |
88
|
imp |
|
90 |
82 84 89
|
syl2anc |
|
91 |
|
simpr |
|
92 |
|
fveq2 |
|
93 |
|
id |
|
94 |
92 93
|
eleq12d |
|
95 |
94
|
rspcv |
|
96 |
95
|
imp |
|
97 |
83 84 96
|
syl2anc |
|
98 |
91 97
|
eqeltrd |
|
99 |
86 93
|
disji |
|
100 |
81 82 83 90 98 99
|
syl122anc |
|
101 |
100
|
ex |
|
102 |
101
|
anasss |
|
103 |
102
|
ralrimivva |
|
104 |
42 103
|
jca |
|
105 |
|
dff13 |
|
106 |
104 105
|
sylibr |
|
107 |
|
f1f1orn |
|
108 |
106 107
|
syl |
|
109 |
|
f1oen3g |
|
110 |
39 108 109
|
sylancr |
|
111 |
110
|
ensymd |
|
112 |
22 23
|
syl |
|
113 |
112
|
ad2antrr |
|
114 |
59
|
ex |
|
115 |
54 114
|
ralrimi |
|
116 |
75
|
ad5antr |
|
117 |
|
simpr |
|
118 |
21
|
ad5antr |
|
119 |
|
simpllr |
|
120 |
118 119
|
sseldd |
|
121 |
|
simplr |
|
122 |
118 121
|
sseldd |
|
123 |
116 117 120 122
|
preimane |
|
124 |
123
|
ex |
|
125 |
124
|
necon4d |
|
126 |
125
|
ralrimiva |
|
127 |
126
|
ex |
|
128 |
54 127
|
ralrimi |
|
129 |
115 128
|
jca |
|
130 |
|
sneq |
|
131 |
130
|
imaeq2d |
|
132 |
1 131
|
f1mpt |
|
133 |
129 132
|
sylibr |
|
134 |
|
f1f1orn |
|
135 |
133 134
|
syl |
|
136 |
|
f1oen3g |
|
137 |
113 135 136
|
syl2anc |
|
138 |
137
|
ensymd |
|
139 |
|
entr |
|
140 |
111 138 139
|
syl2anc |
|
141 |
|
imass2 |
|
142 |
43 141
|
syl |
|
143 |
42 142
|
syl |
|
144 |
|
imauni |
|
145 |
|
imaeq2 |
|
146 |
55
|
adantr |
|
147 |
146 57 58
|
3syl |
|
148 |
145 147
|
iunrnmptss |
|
149 |
|
funimacnv |
|
150 |
75 149
|
syl |
|
151 |
150
|
adantr |
|
152 |
6
|
snssd |
|
153 |
152 5
|
sstrd |
|
154 |
|
df-ss |
|
155 |
153 154
|
sylib |
|
156 |
151 155
|
eqtrd |
|
157 |
156
|
iuneq2dv |
|
158 |
|
iunid |
|
159 |
157 158
|
eqtrdi |
|
160 |
148 159
|
sseqtrd |
|
161 |
160
|
ad2antrr |
|
162 |
144 161
|
eqsstrid |
|
163 |
143 162
|
sstrd |
|
164 |
42
|
adantr |
|
165 |
164
|
ffund |
|
166 |
|
simpr |
|
167 |
55 57
|
syl |
|
168 |
167
|
ad3antrrr |
|
169 |
|
imaexg |
|
170 |
168 169
|
syl |
|
171 |
1 131
|
elrnmpt1s |
|
172 |
166 170 171
|
syl2anc |
|
173 |
164
|
fdmd |
|
174 |
172 173
|
eleqtrrd |
|
175 |
|
fvelrn |
|
176 |
165 174 175
|
syl2anc |
|
177 |
15
|
ad3antrrr |
|
178 |
|
simplr |
|
179 |
|
fveq2 |
|
180 |
|
id |
|
181 |
179 180
|
eleq12d |
|
182 |
181
|
rspcv |
|
183 |
182
|
imp |
|
184 |
172 178 183
|
syl2anc |
|
185 |
|
fniniseg |
|
186 |
185
|
simplbda |
|
187 |
177 184 186
|
syl2anc |
|
188 |
|
fveqeq2 |
|
189 |
188
|
rspcev |
|
190 |
176 187 189
|
syl2anc |
|
191 |
72
|
adantr |
|
192 |
177 191
|
fvelimabd |
|
193 |
190 192
|
mpbird |
|
194 |
193
|
ex |
|
195 |
194
|
ssrdv |
|
196 |
163 195
|
eqssd |
|
197 |
140 196
|
jca |
|
198 |
|
breq1 |
|
199 |
|
imaeq2 |
|
200 |
199
|
eqeq1d |
|
201 |
198 200
|
anbi12d |
|
202 |
201
|
rspcev |
|
203 |
73 197 202
|
syl2anc |
|
204 |
203
|
anasss |
|
205 |
204
|
ex |
|
206 |
205
|
exlimdv |
|
207 |
38 206
|
mpd |
|