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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 5253 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | eeanv 1912 | . . . . . 6 | |
4 | simprll 527 | . . . . . . . . . . 11 | |
5 | simpr 109 | . . . . . . . . . . . . . 14 | |
6 | simpr 109 | . . . . . . . . . . . . . 14 | |
7 | 5, 6 | anim12i 336 | . . . . . . . . . . . . 13 |
8 | funcnveq 5235 | . . . . . . . . . . . . . . . . 17 | |
9 | 8 | biimpi 119 | . . . . . . . . . . . . . . . 16 |
10 | 9 | 19.21bi 1538 | . . . . . . . . . . . . . . 15 |
11 | 10 | 19.21bbi 1539 | . . . . . . . . . . . . . 14 |
12 | 11 | imp 123 | . . . . . . . . . . . . 13 |
13 | 7, 12 | sylan2 284 | . . . . . . . . . . . 12 |
14 | simprrl 529 | . . . . . . . . . . . 12 | |
15 | 13, 14 | eqeltrd 2234 | . . . . . . . . . . 11 |
16 | elin 3291 | . . . . . . . . . . 11 | |
17 | 4, 15, 16 | sylanbrc 414 | . . . . . . . . . 10 |
18 | simprlr 528 | . . . . . . . . . 10 | |
19 | 17, 18 | jca 304 | . . . . . . . . 9 |
20 | 19 | ex 114 | . . . . . . . 8 |
21 | 20 | exlimdv 1799 | . . . . . . 7 |
22 | 21 | eximdv 1860 | . . . . . 6 |
23 | 3, 22 | syl5bir 152 | . . . . 5 |
24 | df-rex 2441 | . . . . . 6 | |
25 | df-rex 2441 | . . . . . 6 | |
26 | 24, 25 | anbi12i 456 | . . . . 5 |
27 | df-rex 2441 | . . . . 5 | |
28 | 23, 26, 27 | 3imtr4g 204 | . . . 4 |
29 | 28 | ss2abdv 3201 | . . 3 |
30 | dfima2 4932 | . . . . 5 | |
31 | dfima2 4932 | . . . . 5 | |
32 | 30, 31 | ineq12i 3307 | . . . 4 |
33 | inab 3376 | . . . 4 | |
34 | 32, 33 | eqtri 2178 | . . 3 |
35 | dfima2 4932 | . . 3 | |
36 | 29, 34, 35 | 3sstr4g 3171 | . 2 |
37 | 2, 36 | eqssd 3145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1333 wceq 1335 wex 1472 wcel 2128 cab 2143 wrex 2436 cin 3101 wss 3102 class class class wbr 3967 ccnv 4587 cima 4591 wfun 5166 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-fun 5174 |
This theorem is referenced by: inpreima 5595 |
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