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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 5174 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | eeanv 1884 | . . . . . 6 | |
4 | simprll 511 | . . . . . . . . . . 11 | |
5 | simpr 109 | . . . . . . . . . . . . . 14 | |
6 | simpr 109 | . . . . . . . . . . . . . 14 | |
7 | 5, 6 | anim12i 336 | . . . . . . . . . . . . 13 |
8 | funcnveq 5156 | . . . . . . . . . . . . . . . . 17 | |
9 | 8 | biimpi 119 | . . . . . . . . . . . . . . . 16 |
10 | 9 | 19.21bi 1522 | . . . . . . . . . . . . . . 15 |
11 | 10 | 19.21bbi 1523 | . . . . . . . . . . . . . 14 |
12 | 11 | imp 123 | . . . . . . . . . . . . 13 |
13 | 7, 12 | sylan2 284 | . . . . . . . . . . . 12 |
14 | simprrl 513 | . . . . . . . . . . . 12 | |
15 | 13, 14 | eqeltrd 2194 | . . . . . . . . . . 11 |
16 | elin 3229 | . . . . . . . . . . 11 | |
17 | 4, 15, 16 | sylanbrc 413 | . . . . . . . . . 10 |
18 | simprlr 512 | . . . . . . . . . 10 | |
19 | 17, 18 | jca 304 | . . . . . . . . 9 |
20 | 19 | ex 114 | . . . . . . . 8 |
21 | 20 | exlimdv 1775 | . . . . . . 7 |
22 | 21 | eximdv 1836 | . . . . . 6 |
23 | 3, 22 | syl5bir 152 | . . . . 5 |
24 | df-rex 2399 | . . . . . 6 | |
25 | df-rex 2399 | . . . . . 6 | |
26 | 24, 25 | anbi12i 455 | . . . . 5 |
27 | df-rex 2399 | . . . . 5 | |
28 | 23, 26, 27 | 3imtr4g 204 | . . . 4 |
29 | 28 | ss2abdv 3140 | . . 3 |
30 | dfima2 4853 | . . . . 5 | |
31 | dfima2 4853 | . . . . 5 | |
32 | 30, 31 | ineq12i 3245 | . . . 4 |
33 | inab 3314 | . . . 4 | |
34 | 32, 33 | eqtri 2138 | . . 3 |
35 | dfima2 4853 | . . 3 | |
36 | 29, 34, 35 | 3sstr4g 3110 | . 2 |
37 | 2, 36 | eqssd 3084 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1314 wceq 1316 wex 1453 wcel 1465 cab 2103 wrex 2394 cin 3040 wss 3041 class class class wbr 3899 ccnv 4508 cima 4512 wfun 5087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-fun 5095 |
This theorem is referenced by: inpreima 5514 |
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