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Mirrors > Home > ILE Home > Th. List > imadif | Unicode version |
Description: The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.) |
Ref | Expression |
---|---|
imadif |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anandir 580 | . . . . . . . 8 | |
2 | 1 | exbii 1584 | . . . . . . 7 |
3 | 19.40 1610 | . . . . . . 7 | |
4 | 2, 3 | sylbi 120 | . . . . . 6 |
5 | nfv 1508 | . . . . . . . . . . 11 | |
6 | nfe1 1472 | . . . . . . . . . . 11 | |
7 | 5, 6 | nfan 1544 | . . . . . . . . . 10 |
8 | funmo 5133 | . . . . . . . . . . . . . 14 | |
9 | vex 2684 | . . . . . . . . . . . . . . . 16 | |
10 | vex 2684 | . . . . . . . . . . . . . . . 16 | |
11 | 9, 10 | brcnv 4717 | . . . . . . . . . . . . . . 15 |
12 | 11 | mobii 2034 | . . . . . . . . . . . . . 14 |
13 | 8, 12 | sylib 121 | . . . . . . . . . . . . 13 |
14 | mopick 2075 | . . . . . . . . . . . . 13 | |
15 | 13, 14 | sylan 281 | . . . . . . . . . . . 12 |
16 | 15 | con2d 613 | . . . . . . . . . . 11 |
17 | imnan 679 | . . . . . . . . . . 11 | |
18 | 16, 17 | sylib 121 | . . . . . . . . . 10 |
19 | 7, 18 | alrimi 1502 | . . . . . . . . 9 |
20 | 19 | ex 114 | . . . . . . . 8 |
21 | exancom 1587 | . . . . . . . 8 | |
22 | alnex 1475 | . . . . . . . 8 | |
23 | 20, 21, 22 | 3imtr3g 203 | . . . . . . 7 |
24 | 23 | anim2d 335 | . . . . . 6 |
25 | 4, 24 | syl5 32 | . . . . 5 |
26 | df-rex 2420 | . . . . . 6 | |
27 | eldif 3075 | . . . . . . . 8 | |
28 | 27 | anbi1i 453 | . . . . . . 7 |
29 | 28 | exbii 1584 | . . . . . 6 |
30 | 26, 29 | bitri 183 | . . . . 5 |
31 | df-rex 2420 | . . . . . 6 | |
32 | df-rex 2420 | . . . . . . 7 | |
33 | 32 | notbii 657 | . . . . . 6 |
34 | 31, 33 | anbi12i 455 | . . . . 5 |
35 | 25, 30, 34 | 3imtr4g 204 | . . . 4 |
36 | 35 | ss2abdv 3165 | . . 3 |
37 | dfima2 4878 | . . 3 | |
38 | dfima2 4878 | . . . . 5 | |
39 | dfima2 4878 | . . . . 5 | |
40 | 38, 39 | difeq12i 3187 | . . . 4 |
41 | difab 3340 | . . . 4 | |
42 | 40, 41 | eqtri 2158 | . . 3 |
43 | 36, 37, 42 | 3sstr4g 3135 | . 2 |
44 | imadiflem 5197 | . . 3 | |
45 | 44 | a1i 9 | . 2 |
46 | 43, 45 | eqssd 3109 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wal 1329 wceq 1331 wex 1468 wcel 1480 wmo 1998 cab 2123 wrex 2415 cdif 3063 wss 3066 class class class wbr 3924 ccnv 4533 cima 4537 wfun 5112 |
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-in1 603 ax-in2 604 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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fun 5120 |
This theorem is referenced by: resdif 5382 difpreima 5540 phplem4 6742 phplem4dom 6749 phplem4on 6754 cnclima 12381 |
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