Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 525 | . . . . . . . 8 | |
2 | 1 | exbii 1496 | . . . . . . 7 |
3 | 19.40 1522 | . . . . . . 7 | |
4 | 2, 3 | sylbi 114 | . . . . . 6 |
5 | nfv 1421 | . . . . . . . . . . 11 | |
6 | nfe1 1385 | . . . . . . . . . . 11 | |
7 | 5, 6 | nfan 1457 | . . . . . . . . . 10 |
8 | funmo 4917 | . . . . . . . . . . . . . 14 | |
9 | vex 2560 | . . . . . . . . . . . . . . . 16 | |
10 | vex 2560 | . . . . . . . . . . . . . . . 16 | |
11 | 9, 10 | brcnv 4518 | . . . . . . . . . . . . . . 15 |
12 | 11 | mobii 1937 | . . . . . . . . . . . . . 14 |
13 | 8, 12 | sylib 127 | . . . . . . . . . . . . 13 |
14 | mopick 1978 | . . . . . . . . . . . . 13 | |
15 | 13, 14 | sylan 267 | . . . . . . . . . . . 12 |
16 | 15 | con2d 554 | . . . . . . . . . . 11 |
17 | imnan 624 | . . . . . . . . . . 11 | |
18 | 16, 17 | sylib 127 | . . . . . . . . . 10 |
19 | 7, 18 | alrimi 1415 | . . . . . . . . 9 |
20 | 19 | ex 108 | . . . . . . . 8 |
21 | exancom 1499 | . . . . . . . 8 | |
22 | alnex 1388 | . . . . . . . 8 | |
23 | 20, 21, 22 | 3imtr3g 193 | . . . . . . 7 |
24 | 23 | anim2d 320 | . . . . . 6 |
25 | 4, 24 | syl5 28 | . . . . 5 |
26 | df-rex 2312 | . . . . . 6 | |
27 | eldif 2927 | . . . . . . . 8 | |
28 | 27 | anbi1i 431 | . . . . . . 7 |
29 | 28 | exbii 1496 | . . . . . 6 |
30 | 26, 29 | bitri 173 | . . . . 5 |
31 | df-rex 2312 | . . . . . 6 | |
32 | df-rex 2312 | . . . . . . 7 | |
33 | 32 | notbii 594 | . . . . . 6 |
34 | 31, 33 | anbi12i 433 | . . . . 5 |
35 | 25, 30, 34 | 3imtr4g 194 | . . . 4 |
36 | 35 | ss2abdv 3013 | . . 3 |
37 | dfima2 4670 | . . 3 | |
38 | dfima2 4670 | . . . . 5 | |
39 | dfima2 4670 | . . . . 5 | |
40 | 38, 39 | difeq12i 3060 | . . . 4 |
41 | difab 3206 | . . . 4 | |
42 | 40, 41 | eqtri 2060 | . . 3 |
43 | 36, 37, 42 | 3sstr4g 2986 | . 2 |
44 | imadiflem 4978 | . . 3 | |
45 | 44 | a1i 9 | . 2 |
46 | 43, 45 | eqssd 2962 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wal 1241 wceq 1243 wex 1381 wcel 1393 wmo 1901 cab 2026 wrex 2307 cdif 2914 wss 2917 class class class wbr 3764 ccnv 4344 cima 4348 wfun 4896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 |
This theorem is referenced by: resdif 5148 difpreima 5294 phplem4 6318 phplem4dom 6324 phplem4on 6329 |
Copyright terms: Public domain | W3C validator |