Step |
Hyp |
Ref |
Expression |
1 |
|
satffunlem2lem1.s |
|
2 |
|
satffunlem2lem1.a |
|
3 |
|
satffunlem2lem1.b |
|
4 |
|
simpl |
|
5 |
4
|
fveq2d |
|
6 |
|
simpr |
|
7 |
6
|
fveq2d |
|
8 |
5 7
|
oveq12d |
|
9 |
8
|
eqeq2d |
|
10 |
4
|
fveq2d |
|
11 |
6
|
fveq2d |
|
12 |
10 11
|
ineq12d |
|
13 |
12
|
difeq2d |
|
14 |
2 13
|
syl5eq |
|
15 |
14
|
eqeq2d |
|
16 |
9 15
|
anbi12d |
|
17 |
16
|
cbvrexdva |
|
18 |
|
simpr |
|
19 |
|
fveq2 |
|
20 |
19
|
adantr |
|
21 |
18 20
|
goaleq12d |
|
22 |
21
|
eqeq2d |
|
23 |
3
|
eqeq2i |
|
24 |
|
opeq1 |
|
25 |
24
|
sneqd |
|
26 |
|
sneq |
|
27 |
26
|
difeq2d |
|
28 |
27
|
reseq2d |
|
29 |
25 28
|
uneq12d |
|
30 |
29
|
adantl |
|
31 |
|
fveq2 |
|
32 |
31
|
adantr |
|
33 |
30 32
|
eleq12d |
|
34 |
33
|
ralbidv |
|
35 |
34
|
rabbidv |
|
36 |
35
|
eqeq2d |
|
37 |
23 36
|
syl5bb |
|
38 |
22 37
|
anbi12d |
|
39 |
38
|
cbvrexdva |
|
40 |
17 39
|
orbi12d |
|
41 |
40
|
cbvrexvw |
|
42 |
|
fveq2 |
|
43 |
19 42
|
oveqan12d |
|
44 |
43
|
eqeq2d |
|
45 |
2
|
eqeq2i |
|
46 |
|
fveq2 |
|
47 |
31 46
|
ineqan12d |
|
48 |
47
|
difeq2d |
|
49 |
48
|
eqeq2d |
|
50 |
45 49
|
syl5bb |
|
51 |
44 50
|
anbi12d |
|
52 |
51
|
cbvrexdva |
|
53 |
52
|
cbvrexvw |
|
54 |
41 53
|
orbi12i |
|
55 |
|
simp-5l |
|
56 |
|
eldifi |
|
57 |
56
|
adantl |
|
58 |
57
|
anim1i |
|
59 |
58
|
ad2antrr |
|
60 |
|
eldifi |
|
61 |
60
|
adantl |
|
62 |
61
|
anim1i |
|
63 |
55 59 62
|
3jca |
|
64 |
|
id |
|
65 |
2
|
eqeq2i |
|
66 |
65
|
biimpi |
|
67 |
66
|
anim2i |
|
68 |
|
satffunlem |
|
69 |
63 64 67 68
|
syl3an |
|
70 |
69
|
3exp |
|
71 |
70
|
com23 |
|
72 |
71
|
rexlimdva |
|
73 |
|
eqeq1 |
|
74 |
|
fvex |
|
75 |
|
fvex |
|
76 |
|
gonafv |
|
77 |
74 75 76
|
mp2an |
|
78 |
|
df-goal |
|
79 |
77 78
|
eqeq12i |
|
80 |
|
1oex |
|
81 |
|
opex |
|
82 |
80 81
|
opth |
|
83 |
|
1one2o |
|
84 |
|
df-ne |
|
85 |
|
pm2.21 |
|
86 |
84 85
|
sylbi |
|
87 |
83 86
|
ax-mp |
|
88 |
87
|
adantr |
|
89 |
82 88
|
sylbi |
|
90 |
79 89
|
sylbi |
|
91 |
73 90
|
syl6bi |
|
92 |
91
|
adantr |
|
93 |
92
|
com12 |
|
94 |
93
|
adantr |
|
95 |
94
|
a1i |
|
96 |
95
|
rexlimdva |
|
97 |
72 96
|
jaod |
|
98 |
97
|
rexlimdva |
|
99 |
|
simp-4l |
|
100 |
58
|
adantr |
|
101 |
|
ssel |
|
102 |
101
|
ad3antlr |
|
103 |
102
|
com12 |
|
104 |
103
|
adantr |
|
105 |
104
|
impcom |
|
106 |
|
eldifi |
|
107 |
106
|
ad2antll |
|
108 |
105 107
|
jca |
|
109 |
99 100 108
|
3jca |
|
110 |
109 64 67 68
|
syl3an |
|
111 |
110
|
3exp |
|
112 |
111
|
com23 |
|
113 |
112
|
rexlimdvva |
|
114 |
98 113
|
jaod |
|
115 |
114
|
com23 |
|
116 |
115
|
rexlimdva |
|
117 |
|
eqeq1 |
|
118 |
|
df-goal |
|
119 |
|
fvex |
|
120 |
|
fvex |
|
121 |
|
gonafv |
|
122 |
119 120 121
|
mp2an |
|
123 |
118 122
|
eqeq12i |
|
124 |
|
2oex |
|
125 |
|
opex |
|
126 |
124 125
|
opth |
|
127 |
87
|
eqcoms |
|
128 |
127
|
adantr |
|
129 |
126 128
|
sylbi |
|
130 |
123 129
|
sylbi |
|
131 |
117 130
|
syl6bi |
|
132 |
131
|
adantr |
|
133 |
132
|
com12 |
|
134 |
133
|
adantr |
|
135 |
134
|
rexlimivw |
|
136 |
135
|
a1i |
|
137 |
|
eqeq1 |
|
138 |
78 118
|
eqeq12i |
|
139 |
|
opex |
|
140 |
124 139
|
opth |
|
141 |
|
vex |
|
142 |
141 119
|
opth |
|
143 |
142
|
anbi2i |
|
144 |
138 140 143
|
3bitri |
|
145 |
137 144
|
bitrdi |
|
146 |
145
|
adantl |
|
147 |
56
|
a1i |
|
148 |
|
funfv1st2nd |
|
149 |
148
|
ex |
|
150 |
149
|
adantr |
|
151 |
|
funfv1st2nd |
|
152 |
151
|
ex |
|
153 |
152
|
adantr |
|
154 |
|
fveqeq2 |
|
155 |
|
eqtr2 |
|
156 |
29
|
eqcomd |
|
157 |
156
|
adantl |
|
158 |
|
simpl |
|
159 |
158
|
eqcomd |
|
160 |
157 159
|
eleq12d |
|
161 |
160
|
ralbidv |
|
162 |
161
|
rabbidv |
|
163 |
162 3
|
eqtr4di |
|
164 |
|
eqeq12 |
|
165 |
163 164
|
syl5ibrcom |
|
166 |
165
|
exp4b |
|
167 |
155 166
|
syl |
|
168 |
167
|
ex |
|
169 |
154 168
|
syl6bi |
|
170 |
169
|
com24 |
|
171 |
170
|
impcom |
|
172 |
171
|
com13 |
|
173 |
60 153 172
|
syl56 |
|
174 |
173
|
com23 |
|
175 |
147 150 174
|
3syld |
|
176 |
175
|
imp |
|
177 |
176
|
adantr |
|
178 |
177
|
imp |
|
179 |
178
|
adantld |
|
180 |
179
|
ad2antrr |
|
181 |
146 180
|
sylbid |
|
182 |
181
|
impd |
|
183 |
182
|
ex |
|
184 |
183
|
com34 |
|
185 |
184
|
impd |
|
186 |
185
|
rexlimdva |
|
187 |
136 186
|
jaod |
|
188 |
187
|
rexlimdva |
|
189 |
134
|
a1i |
|
190 |
189
|
rexlimdva |
|
191 |
190
|
rexlimdva |
|
192 |
188 191
|
jaod |
|
193 |
192
|
com23 |
|
194 |
193
|
rexlimdva |
|
195 |
116 194
|
jaod |
|
196 |
195
|
rexlimdva |
|
197 |
|
simplll |
|
198 |
|
ssel |
|
199 |
198
|
adantrd |
|
200 |
199
|
adantl |
|
201 |
200
|
imp |
|
202 |
|
eldifi |
|
203 |
202
|
ad2antll |
|
204 |
201 203
|
jca |
|
205 |
204
|
adantr |
|
206 |
60
|
anim1i |
|
207 |
206
|
adantl |
|
208 |
197 205 207
|
3jca |
|
209 |
208
|
adantr |
|
210 |
|
simprl |
|
211 |
67
|
ad2antll |
|
212 |
209 210 211 68
|
syl3anc |
|
213 |
212
|
exp32 |
|
214 |
213
|
impancom |
|
215 |
214
|
expdimp |
|
216 |
215
|
rexlimdv |
|
217 |
91
|
adantrd |
|
218 |
217
|
adantr |
|
219 |
218
|
ad3antlr |
|
220 |
219
|
rexlimdva |
|
221 |
216 220
|
jaod |
|
222 |
221
|
rexlimdva |
|
223 |
|
simplll |
|
224 |
204
|
adantr |
|
225 |
101
|
adantl |
|
226 |
225
|
adantr |
|
227 |
226
|
com12 |
|
228 |
227
|
adantr |
|
229 |
228
|
impcom |
|
230 |
106
|
ad2antll |
|
231 |
229 230
|
jca |
|
232 |
223 224 231
|
3jca |
|
233 |
232 64 67 68
|
syl3an |
|
234 |
233
|
3exp |
|
235 |
234
|
impancom |
|
236 |
235
|
rexlimdvv |
|
237 |
222 236
|
jaod |
|
238 |
237
|
ex |
|
239 |
238
|
rexlimdvva |
|
240 |
196 239
|
jaod |
|
241 |
54 240
|
syl5bi |
|
242 |
241
|
impd |
|
243 |
242
|
alrimivv |
|
244 |
|
eqeq1 |
|
245 |
244
|
anbi2d |
|
246 |
245
|
rexbidv |
|
247 |
|
eqeq1 |
|
248 |
247
|
anbi2d |
|
249 |
248
|
rexbidv |
|
250 |
246 249
|
orbi12d |
|
251 |
250
|
rexbidv |
|
252 |
245
|
2rexbidv |
|
253 |
251 252
|
orbi12d |
|
254 |
253
|
mo4 |
|
255 |
243 254
|
sylibr |
|
256 |
255
|
alrimiv |
|
257 |
|
funopab |
|
258 |
256 257
|
sylibr |
|