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Mirrors > Home > NFE Home > Th. List > imagekexg | Unicode version |
Description: The Kuratowski image functor preserves sethood. (Contributed by SF, 14-Jan-2015.) |
Ref | Expression |
---|---|
imagekexg | Imagek |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-imagek 4194 | . 2 Imagek k Ins2k Sk Ins3k Sk k k SIk k1 1 1c | |
2 | sikexg 4296 | . . . . . . . 8 SIk | |
3 | cnvkexg 4286 | . . . . . . . 8 SIk k SIk | |
4 | 2, 3 | syl 15 | . . . . . . 7 k SIk |
5 | ssetkex 4294 | . . . . . . . 8 Sk | |
6 | cokexg 4309 | . . . . . . . 8 Sk k SIk Sk k k SIk | |
7 | 5, 6 | mpan 651 | . . . . . . 7 k SIk Sk k k SIk |
8 | 4, 7 | syl 15 | . . . . . 6 Sk k k SIk |
9 | ins3kexg 4306 | . . . . . 6 Sk k k SIk Ins3k Sk k k SIk | |
10 | 8, 9 | syl 15 | . . . . 5 Ins3k Sk k k SIk |
11 | 5 | ins2kex 4307 | . . . . . 6 Ins2k Sk |
12 | symdifexg 4103 | . . . . . 6 Ins2k Sk Ins3k Sk k k SIk Ins2k Sk Ins3k Sk k k SIk | |
13 | 11, 12 | mpan 651 | . . . . 5 Ins3k Sk k k SIk Ins2k Sk Ins3k Sk k k SIk |
14 | 10, 13 | syl 15 | . . . 4 Ins2k Sk Ins3k Sk k k SIk |
15 | 1cex 4142 | . . . . . . 7 1c | |
16 | 15 | pw1ex 4303 | . . . . . 6 1 1c |
17 | 16 | pw1ex 4303 | . . . . 5 1 1 1c |
18 | imakexg 4299 | . . . . 5 Ins2k Sk Ins3k Sk k k SIk 1 1 1c Ins2k Sk Ins3k Sk k k SIk k1 1 1c | |
19 | 17, 18 | mpan2 652 | . . . 4 Ins2k Sk Ins3k Sk k k SIk Ins2k Sk Ins3k Sk k k SIk k1 1 1c |
20 | 14, 19 | syl 15 | . . 3 Ins2k Sk Ins3k Sk k k SIk k1 1 1c |
21 | vvex 4109 | . . . . 5 | |
22 | 21, 21 | xpkex 4289 | . . . 4 k |
23 | difexg 4102 | . . . 4 k Ins2k Sk Ins3k Sk k k SIk k1 1 1c k Ins2k Sk Ins3k Sk k k SIk k1 1 1c | |
24 | 22, 23 | mpan 651 | . . 3 Ins2k Sk Ins3k Sk k k SIk k1 1 1c k Ins2k Sk Ins3k Sk k k SIk k1 1 1c |
25 | 20, 24 | syl 15 | . 2 k Ins2k Sk Ins3k Sk k k SIk k1 1 1c |
26 | 1, 25 | syl5eqel 2437 | 1 Imagek |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wcel 1710 cvv 2859 cdif 3206 csymdif 3209 1cc1c 4134 1 cpw1 4135 k cxpk 4174 kccnvk 4175 Ins2k cins2k 4176 Ins3k cins3k 4177 kcimak 4179 k ccomk 4180 SIk csik 4181 Imagekcimagek 4182 Sk cssetk 4183 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-pr 3742 df-opk 4058 df-1c 4136 df-pw1 4137 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 |
This theorem is referenced by: imagekex 4312 |
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