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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 5279 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | eeanv 1925 | . . . . . 6 | |
4 | simprll 532 | . . . . . . . . . . 11 | |
5 | simpr 109 | . . . . . . . . . . . . . 14 | |
6 | simpr 109 | . . . . . . . . . . . . . 14 | |
7 | 5, 6 | anim12i 336 | . . . . . . . . . . . . 13 |
8 | funcnveq 5261 | . . . . . . . . . . . . . . . . 17 | |
9 | 8 | biimpi 119 | . . . . . . . . . . . . . . . 16 |
10 | 9 | 19.21bi 1551 | . . . . . . . . . . . . . . 15 |
11 | 10 | 19.21bbi 1552 | . . . . . . . . . . . . . 14 |
12 | 11 | imp 123 | . . . . . . . . . . . . 13 |
13 | 7, 12 | sylan2 284 | . . . . . . . . . . . 12 |
14 | simprrl 534 | . . . . . . . . . . . 12 | |
15 | 13, 14 | eqeltrd 2247 | . . . . . . . . . . 11 |
16 | elin 3310 | . . . . . . . . . . 11 | |
17 | 4, 15, 16 | sylanbrc 415 | . . . . . . . . . 10 |
18 | simprlr 533 | . . . . . . . . . 10 | |
19 | 17, 18 | jca 304 | . . . . . . . . 9 |
20 | 19 | ex 114 | . . . . . . . 8 |
21 | 20 | exlimdv 1812 | . . . . . . 7 |
22 | 21 | eximdv 1873 | . . . . . 6 |
23 | 3, 22 | syl5bir 152 | . . . . 5 |
24 | df-rex 2454 | . . . . . 6 | |
25 | df-rex 2454 | . . . . . 6 | |
26 | 24, 25 | anbi12i 457 | . . . . 5 |
27 | df-rex 2454 | . . . . 5 | |
28 | 23, 26, 27 | 3imtr4g 204 | . . . 4 |
29 | 28 | ss2abdv 3220 | . . 3 |
30 | dfima2 4955 | . . . . 5 | |
31 | dfima2 4955 | . . . . 5 | |
32 | 30, 31 | ineq12i 3326 | . . . 4 |
33 | inab 3395 | . . . 4 | |
34 | 32, 33 | eqtri 2191 | . . 3 |
35 | dfima2 4955 | . . 3 | |
36 | 29, 34, 35 | 3sstr4g 3190 | . 2 |
37 | 2, 36 | eqssd 3164 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1346 wceq 1348 wex 1485 wcel 2141 cab 2156 wrex 2449 cin 3120 wss 3121 class class class wbr 3989 ccnv 4610 cima 4614 wfun 5192 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-fun 5200 |
This theorem is referenced by: inpreima 5622 |
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