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Mirrors > Home > ILE Home > Th. List > imain | Unicode version |
Description: The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.) |
Ref | Expression |
---|---|
imain |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imainlem 4980 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | eeanv 1807 | . . . . . 6 | |
4 | simprll 489 | . . . . . . . . . . 11 | |
5 | simpr 103 | . . . . . . . . . . . . . 14 | |
6 | simpr 103 | . . . . . . . . . . . . . 14 | |
7 | 5, 6 | anim12i 321 | . . . . . . . . . . . . 13 |
8 | funcnveq 4962 | . . . . . . . . . . . . . . . . 17 | |
9 | 8 | biimpi 113 | . . . . . . . . . . . . . . . 16 |
10 | 9 | 19.21bi 1450 | . . . . . . . . . . . . . . 15 |
11 | 10 | 19.21bbi 1451 | . . . . . . . . . . . . . 14 |
12 | 11 | imp 115 | . . . . . . . . . . . . 13 |
13 | 7, 12 | sylan2 270 | . . . . . . . . . . . 12 |
14 | simprrl 491 | . . . . . . . . . . . 12 | |
15 | 13, 14 | eqeltrd 2114 | . . . . . . . . . . 11 |
16 | elin 3126 | . . . . . . . . . . 11 | |
17 | 4, 15, 16 | sylanbrc 394 | . . . . . . . . . 10 |
18 | simprlr 490 | . . . . . . . . . 10 | |
19 | 17, 18 | jca 290 | . . . . . . . . 9 |
20 | 19 | ex 108 | . . . . . . . 8 |
21 | 20 | exlimdv 1700 | . . . . . . 7 |
22 | 21 | eximdv 1760 | . . . . . 6 |
23 | 3, 22 | syl5bir 142 | . . . . 5 |
24 | df-rex 2312 | . . . . . 6 | |
25 | df-rex 2312 | . . . . . 6 | |
26 | 24, 25 | anbi12i 433 | . . . . 5 |
27 | df-rex 2312 | . . . . 5 | |
28 | 23, 26, 27 | 3imtr4g 194 | . . . 4 |
29 | 28 | ss2abdv 3013 | . . 3 |
30 | dfima2 4670 | . . . . 5 | |
31 | dfima2 4670 | . . . . 5 | |
32 | 30, 31 | ineq12i 3136 | . . . 4 |
33 | inab 3205 | . . . 4 | |
34 | 32, 33 | eqtri 2060 | . . 3 |
35 | dfima2 4670 | . . 3 | |
36 | 29, 34, 35 | 3sstr4g 2986 | . 2 |
37 | 2, 36 | eqssd 2962 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wal 1241 wceq 1243 wex 1381 wcel 1393 cab 2026 wrex 2307 cin 2916 wss 2917 class class class wbr 3764 ccnv 4344 cima 4348 wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 |
This theorem is referenced by: inpreima 5293 |
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