Step |
Hyp |
Ref |
Expression |
1 |
|
nllytop |
|
2 |
|
elinel1 |
|
3 |
|
kgentop |
|
4 |
2 3
|
syl |
|
5 |
|
txtop |
|
6 |
1 4 5
|
syl2an |
|
7 |
|
simplll |
|
8 |
|
eqid |
|
9 |
8
|
mptpreima |
|
10 |
1
|
ad3antrrr |
|
11 |
|
toptopon2 |
|
12 |
10 11
|
sylib |
|
13 |
|
idcn |
|
14 |
12 13
|
syl |
|
15 |
|
simpllr |
|
16 |
15 4
|
syl |
|
17 |
|
toptopon2 |
|
18 |
16 17
|
sylib |
|
19 |
|
simpr |
|
20 |
|
simplr |
|
21 |
|
elunii |
|
22 |
19 20 21
|
syl2anc |
|
23 |
|
eqid |
|
24 |
|
eqid |
|
25 |
23 24
|
txuni |
|
26 |
10 16 25
|
syl2anc |
|
27 |
10 16 5
|
syl2anc |
|
28 |
|
eqid |
|
29 |
28
|
kgenuni |
|
30 |
27 29
|
syl |
|
31 |
26 30
|
eqtrd |
|
32 |
22 31
|
eleqtrrd |
|
33 |
|
xp2nd |
|
34 |
32 33
|
syl |
|
35 |
|
cnconst2 |
|
36 |
12 18 34 35
|
syl3anc |
|
37 |
|
fvresi |
|
38 |
|
fvex |
|
39 |
38
|
fvconst2 |
|
40 |
37 39
|
opeq12d |
|
41 |
40
|
mpteq2ia |
|
42 |
41
|
eqcomi |
|
43 |
23 42
|
txcnmpt |
|
44 |
14 36 43
|
syl2anc |
|
45 |
|
llycmpkgen |
|
46 |
45
|
ad3antrrr |
|
47 |
6
|
ad2antrr |
|
48 |
|
kgencn3 |
|
49 |
46 47 48
|
syl2anc |
|
50 |
44 49
|
eleqtrd |
|
51 |
|
cnima |
|
52 |
50 20 51
|
syl2anc |
|
53 |
9 52
|
eqeltrrid |
|
54 |
|
opeq1 |
|
55 |
54
|
eleq1d |
|
56 |
|
xp1st |
|
57 |
32 56
|
syl |
|
58 |
|
1st2nd2 |
|
59 |
32 58
|
syl |
|
60 |
59 19
|
eqeltrrd |
|
61 |
55 57 60
|
elrabd |
|
62 |
|
nlly2i |
|
63 |
7 53 61 62
|
syl3anc |
|
64 |
10
|
adantr |
|
65 |
16
|
adantr |
|
66 |
|
simprlr |
|
67 |
|
ssrab2 |
|
68 |
67
|
a1i |
|
69 |
|
incom |
|
70 |
|
simprll |
|
71 |
70
|
elpwid |
|
72 |
|
ssrab2 |
|
73 |
71 72
|
sstrdi |
|
74 |
73
|
adantr |
|
75 |
|
elpwi |
|
76 |
75
|
ad2antrl |
|
77 |
|
eldif |
|
78 |
77
|
anbi1i |
|
79 |
|
anass |
|
80 |
78 79
|
bitri |
|
81 |
80
|
rexbii2 |
|
82 |
|
ancom |
|
83 |
|
fveqeq2 |
|
84 |
|
eleq1 |
|
85 |
84
|
notbid |
|
86 |
83 85
|
anbi12d |
|
87 |
82 86
|
syl5bb |
|
88 |
87
|
rexxp |
|
89 |
81 88
|
bitri |
|
90 |
|
simpl |
|
91 |
90
|
sselda |
|
92 |
91
|
adantr |
|
93 |
|
simplr |
|
94 |
93
|
sselda |
|
95 |
92 94
|
opelxpd |
|
96 |
95
|
fvresd |
|
97 |
|
vex |
|
98 |
|
vex |
|
99 |
97 98
|
op2nd |
|
100 |
96 99
|
eqtrdi |
|
101 |
100
|
eqeq1d |
|
102 |
101
|
anbi1d |
|
103 |
102
|
rexbidva |
|
104 |
|
opeq2 |
|
105 |
104
|
eleq1d |
|
106 |
105
|
notbid |
|
107 |
106
|
ceqsrexbv |
|
108 |
103 107
|
bitrdi |
|
109 |
108
|
rexbidva |
|
110 |
|
r19.42v |
|
111 |
109 110
|
bitrdi |
|
112 |
89 111
|
syl5bb |
|
113 |
|
f2ndres |
|
114 |
|
ffn |
|
115 |
113 114
|
ax-mp |
|
116 |
|
difss |
|
117 |
|
xpss12 |
|
118 |
116 117
|
sstrid |
|
119 |
|
fvelimab |
|
120 |
115 118 119
|
sylancr |
|
121 |
|
eldif |
|
122 |
|
simpr |
|
123 |
122
|
sselda |
|
124 |
|
sneq |
|
125 |
124
|
xpeq2d |
|
126 |
125
|
sseq1d |
|
127 |
|
dfss3 |
|
128 |
|
eleq1 |
|
129 |
128
|
ralxp |
|
130 |
|
vex |
|
131 |
|
opeq2 |
|
132 |
131
|
eleq1d |
|
133 |
130 132
|
ralsn |
|
134 |
133
|
ralbii |
|
135 |
127 129 134
|
3bitri |
|
136 |
126 135
|
bitrdi |
|
137 |
136
|
elrab3 |
|
138 |
123 137
|
syl |
|
139 |
138
|
notbid |
|
140 |
|
rexnal |
|
141 |
139 140
|
bitr4di |
|
142 |
141
|
pm5.32da |
|
143 |
121 142
|
syl5bb |
|
144 |
112 120 143
|
3bitr4d |
|
145 |
144
|
eqrdv |
|
146 |
74 76 145
|
syl2anc |
|
147 |
|
difin |
|
148 |
65
|
adantr |
|
149 |
24
|
restuni |
|
150 |
148 76 149
|
syl2anc |
|
151 |
150
|
difeq1d |
|
152 |
147 151
|
syl5eqr |
|
153 |
146 152
|
eqtrd |
|
154 |
15
|
ad2antrr |
|
155 |
154
|
elin2d |
|
156 |
|
df-ima |
|
157 |
|
resres |
|
158 |
|
inss2 |
|
159 |
158 116
|
sstri |
|
160 |
|
ssres2 |
|
161 |
159 160
|
ax-mp |
|
162 |
157 161
|
eqsstri |
|
163 |
162
|
rnssi |
|
164 |
156 163
|
eqsstri |
|
165 |
|
f2ndres |
|
166 |
|
frn |
|
167 |
165 166
|
ax-mp |
|
168 |
164 167
|
sstri |
|
169 |
168 76
|
sstrid |
|
170 |
12
|
ad2antrr |
|
171 |
148 17
|
sylib |
|
172 |
|
tx2cn |
|
173 |
170 171 172
|
syl2anc |
|
174 |
27
|
ad2antrr |
|
175 |
116
|
a1i |
|
176 |
|
vex |
|
177 |
|
vex |
|
178 |
176 177
|
xpex |
|
179 |
178
|
a1i |
|
180 |
|
restabs |
|
181 |
174 175 179 180
|
syl3anc |
|
182 |
64
|
adantr |
|
183 |
154 4
|
syl |
|
184 |
176
|
a1i |
|
185 |
|
simprl |
|
186 |
|
txrest |
|
187 |
182 183 184 185 186
|
syl22anc |
|
188 |
|
simprr3 |
|
189 |
188
|
adantr |
|
190 |
|
simprr |
|
191 |
|
txcmp |
|
192 |
189 190 191
|
syl2anc |
|
193 |
187 192
|
eqeltrd |
|
194 |
|
difin |
|
195 |
74 76 117
|
syl2anc |
|
196 |
182 148 25
|
syl2anc |
|
197 |
195 196
|
sseqtrd |
|
198 |
28
|
restuni |
|
199 |
174 197 198
|
syl2anc |
|
200 |
199
|
difeq1d |
|
201 |
194 200
|
syl5eqr |
|
202 |
|
resttop |
|
203 |
174 178 202
|
sylancl |
|
204 |
|
incom |
|
205 |
20
|
ad2antrr |
|
206 |
|
kgeni |
|
207 |
205 193 206
|
syl2anc |
|
208 |
204 207
|
eqeltrid |
|
209 |
|
eqid |
|
210 |
209
|
opncld |
|
211 |
203 208 210
|
syl2anc |
|
212 |
201 211
|
eqeltrd |
|
213 |
|
cmpcld |
|
214 |
193 212 213
|
syl2anc |
|
215 |
181 214
|
eqeltrrd |
|
216 |
|
imacmp |
|
217 |
173 215 216
|
syl2anc |
|
218 |
24
|
hauscmp |
|
219 |
155 169 217 218
|
syl3anc |
|
220 |
168
|
a1i |
|
221 |
24
|
restcldi |
|
222 |
76 219 220 221
|
syl3anc |
|
223 |
153 222
|
eqeltrrd |
|
224 |
|
resttop |
|
225 |
148 185 224
|
syl2anc |
|
226 |
|
inss1 |
|
227 |
226 150
|
sseqtrid |
|
228 |
|
eqid |
|
229 |
228
|
isopn2 |
|
230 |
225 227 229
|
syl2anc |
|
231 |
223 230
|
mpbird |
|
232 |
69 231
|
eqeltrid |
|
233 |
232
|
expr |
|
234 |
233
|
ralrimiva |
|
235 |
65 17
|
sylib |
|
236 |
|
elkgen |
|
237 |
235 236
|
syl |
|
238 |
68 234 237
|
mpbir2and |
|
239 |
15
|
adantr |
|
240 |
239 2
|
syl |
|
241 |
|
kgenidm |
|
242 |
240 241
|
syl |
|
243 |
238 242
|
eleqtrd |
|
244 |
|
txopn |
|
245 |
64 65 66 243 244
|
syl22anc |
|
246 |
59
|
adantr |
|
247 |
|
simprr1 |
|
248 |
|
sneq |
|
249 |
248
|
xpeq2d |
|
250 |
249
|
sseq1d |
|
251 |
34
|
adantr |
|
252 |
|
relxp |
|
253 |
252
|
a1i |
|
254 |
|
opelxp |
|
255 |
71
|
sselda |
|
256 |
|
opeq1 |
|
257 |
256
|
eleq1d |
|
258 |
257
|
elrab |
|
259 |
258
|
simprbi |
|
260 |
255 259
|
syl |
|
261 |
|
elsni |
|
262 |
261
|
opeq2d |
|
263 |
262
|
eleq1d |
|
264 |
260 263
|
syl5ibrcom |
|
265 |
264
|
expimpd |
|
266 |
254 265
|
syl5bi |
|
267 |
253 266
|
relssdv |
|
268 |
250 251 267
|
elrabd |
|
269 |
247 268
|
opelxpd |
|
270 |
246 269
|
eqeltrd |
|
271 |
|
relxp |
|
272 |
271
|
a1i |
|
273 |
|
opelxp |
|
274 |
126
|
elrab |
|
275 |
274
|
simprbi |
|
276 |
|
simprr2 |
|
277 |
276
|
sselda |
|
278 |
|
vsnid |
|
279 |
|
opelxpi |
|
280 |
277 278 279
|
sylancl |
|
281 |
|
ssel |
|
282 |
275 280 281
|
syl2imc |
|
283 |
282
|
expimpd |
|
284 |
273 283
|
syl5bi |
|
285 |
272 284
|
relssdv |
|
286 |
|
eleq2 |
|
287 |
|
sseq1 |
|
288 |
286 287
|
anbi12d |
|
289 |
288
|
rspcev |
|
290 |
245 270 285 289
|
syl12anc |
|
291 |
290
|
expr |
|
292 |
291
|
rexlimdvva |
|
293 |
63 292
|
mpd |
|
294 |
293
|
ralrimiva |
|
295 |
6
|
adantr |
|
296 |
|
eltop2 |
|
297 |
295 296
|
syl |
|
298 |
294 297
|
mpbird |
|
299 |
298
|
ex |
|
300 |
299
|
ssrdv |
|
301 |
|
iskgen2 |
|
302 |
6 300 301
|
sylanbrc |
|