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Mirrors > Home > ILE Home > Th. List > cnviinm | Unicode version |
Description: The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Ref | Expression |
---|---|
cnviinm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2178 | . . 3 | |
2 | 1 | cbvexv 1872 | . 2 |
3 | eleq1w 2178 | . . . 4 | |
4 | 3 | cbvexv 1872 | . . 3 |
5 | relcnv 4887 | . . . 4 | |
6 | r19.2m 3419 | . . . . . . . 8 | |
7 | 6 | expcom 115 | . . . . . . 7 |
8 | relcnv 4887 | . . . . . . . . 9 | |
9 | df-rel 4516 | . . . . . . . . 9 | |
10 | 8, 9 | mpbi 144 | . . . . . . . 8 |
11 | 10 | a1i 9 | . . . . . . 7 |
12 | 7, 11 | mprg 2466 | . . . . . 6 |
13 | iinss 3834 | . . . . . 6 | |
14 | 12, 13 | syl 14 | . . . . 5 |
15 | df-rel 4516 | . . . . 5 | |
16 | 14, 15 | sylibr 133 | . . . 4 |
17 | vex 2663 | . . . . . . . 8 | |
18 | vex 2663 | . . . . . . . 8 | |
19 | 17, 18 | opex 4121 | . . . . . . 7 |
20 | eliin 3788 | . . . . . . 7 | |
21 | 19, 20 | ax-mp 5 | . . . . . 6 |
22 | 18, 17 | opelcnv 4691 | . . . . . 6 |
23 | 18, 17 | opex 4121 | . . . . . . . 8 |
24 | eliin 3788 | . . . . . . . 8 | |
25 | 23, 24 | ax-mp 5 | . . . . . . 7 |
26 | 18, 17 | opelcnv 4691 | . . . . . . . 8 |
27 | 26 | ralbii 2418 | . . . . . . 7 |
28 | 25, 27 | bitri 183 | . . . . . 6 |
29 | 21, 22, 28 | 3bitr4i 211 | . . . . 5 |
30 | 29 | eqrelriv 4602 | . . . 4 |
31 | 5, 16, 30 | sylancr 410 | . . 3 |
32 | 4, 31 | sylbir 134 | . 2 |
33 | 2, 32 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1316 wex 1453 wcel 1465 wral 2393 wrex 2394 cvv 2660 wss 3041 cop 3500 ciin 3784 cxp 4507 ccnv 4508 wrel 4514 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-iin 3786 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 |
This theorem is referenced by: (None) |
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