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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 5000 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | eeanv 1848 | . . . . . 6 | |
4 | simprll 503 | . . . . . . . . . . 11 | |
5 | simpr 108 | . . . . . . . . . . . . . 14 | |
6 | simpr 108 | . . . . . . . . . . . . . 14 | |
7 | 5, 6 | anim12i 331 | . . . . . . . . . . . . 13 |
8 | funcnveq 4982 | . . . . . . . . . . . . . . . . 17 | |
9 | 8 | biimpi 118 | . . . . . . . . . . . . . . . 16 |
10 | 9 | 19.21bi 1490 | . . . . . . . . . . . . . . 15 |
11 | 10 | 19.21bbi 1491 | . . . . . . . . . . . . . 14 |
12 | 11 | imp 122 | . . . . . . . . . . . . 13 |
13 | 7, 12 | sylan2 280 | . . . . . . . . . . . 12 |
14 | simprrl 505 | . . . . . . . . . . . 12 | |
15 | 13, 14 | eqeltrd 2155 | . . . . . . . . . . 11 |
16 | elin 3155 | . . . . . . . . . . 11 | |
17 | 4, 15, 16 | sylanbrc 408 | . . . . . . . . . 10 |
18 | simprlr 504 | . . . . . . . . . 10 | |
19 | 17, 18 | jca 300 | . . . . . . . . 9 |
20 | 19 | ex 113 | . . . . . . . 8 |
21 | 20 | exlimdv 1740 | . . . . . . 7 |
22 | 21 | eximdv 1801 | . . . . . 6 |
23 | 3, 22 | syl5bir 151 | . . . . 5 |
24 | df-rex 2354 | . . . . . 6 | |
25 | df-rex 2354 | . . . . . 6 | |
26 | 24, 25 | anbi12i 447 | . . . . 5 |
27 | df-rex 2354 | . . . . 5 | |
28 | 23, 26, 27 | 3imtr4g 203 | . . . 4 |
29 | 28 | ss2abdv 3067 | . . 3 |
30 | dfima2 4690 | . . . . 5 | |
31 | dfima2 4690 | . . . . 5 | |
32 | 30, 31 | ineq12i 3165 | . . . 4 |
33 | inab 3232 | . . . 4 | |
34 | 32, 33 | eqtri 2101 | . . 3 |
35 | dfima2 4690 | . . 3 | |
36 | 29, 34, 35 | 3sstr4g 3040 | . 2 |
37 | 2, 36 | eqssd 3016 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wal 1282 wceq 1284 wex 1421 wcel 1433 cab 2067 wrex 2349 cin 2972 wss 2973 class class class wbr 3785 ccnv 4362 cima 4366 wfun 4916 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-fun 4924 |
This theorem is referenced by: inpreima 5314 |
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