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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 5204 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | eeanv 1904 | . . . . . 6 | |
4 | simprll 526 | . . . . . . . . . . 11 | |
5 | simpr 109 | . . . . . . . . . . . . . 14 | |
6 | simpr 109 | . . . . . . . . . . . . . 14 | |
7 | 5, 6 | anim12i 336 | . . . . . . . . . . . . 13 |
8 | funcnveq 5186 | . . . . . . . . . . . . . . . . 17 | |
9 | 8 | biimpi 119 | . . . . . . . . . . . . . . . 16 |
10 | 9 | 19.21bi 1537 | . . . . . . . . . . . . . . 15 |
11 | 10 | 19.21bbi 1538 | . . . . . . . . . . . . . 14 |
12 | 11 | imp 123 | . . . . . . . . . . . . 13 |
13 | 7, 12 | sylan2 284 | . . . . . . . . . . . 12 |
14 | simprrl 528 | . . . . . . . . . . . 12 | |
15 | 13, 14 | eqeltrd 2216 | . . . . . . . . . . 11 |
16 | elin 3259 | . . . . . . . . . . 11 | |
17 | 4, 15, 16 | sylanbrc 413 | . . . . . . . . . 10 |
18 | simprlr 527 | . . . . . . . . . 10 | |
19 | 17, 18 | jca 304 | . . . . . . . . 9 |
20 | 19 | ex 114 | . . . . . . . 8 |
21 | 20 | exlimdv 1791 | . . . . . . 7 |
22 | 21 | eximdv 1852 | . . . . . 6 |
23 | 3, 22 | syl5bir 152 | . . . . 5 |
24 | df-rex 2422 | . . . . . 6 | |
25 | df-rex 2422 | . . . . . 6 | |
26 | 24, 25 | anbi12i 455 | . . . . 5 |
27 | df-rex 2422 | . . . . 5 | |
28 | 23, 26, 27 | 3imtr4g 204 | . . . 4 |
29 | 28 | ss2abdv 3170 | . . 3 |
30 | dfima2 4883 | . . . . 5 | |
31 | dfima2 4883 | . . . . 5 | |
32 | 30, 31 | ineq12i 3275 | . . . 4 |
33 | inab 3344 | . . . 4 | |
34 | 32, 33 | eqtri 2160 | . . 3 |
35 | dfima2 4883 | . . 3 | |
36 | 29, 34, 35 | 3sstr4g 3140 | . 2 |
37 | 2, 36 | eqssd 3114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 wceq 1331 wex 1468 wcel 1480 cab 2125 wrex 2417 cin 3070 wss 3071 class class class wbr 3929 ccnv 4538 cima 4542 wfun 5117 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 |
This theorem is referenced by: inpreima 5546 |
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