Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xpiindim | Unicode version |
Description: Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.) |
Ref | Expression |
---|---|
xpiindim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 4618 | . . . . . 6 | |
2 | 1 | rgenw 2464 | . . . . 5 |
3 | r19.2m 3419 | . . . . 5 | |
4 | 2, 3 | mpan2 421 | . . . 4 |
5 | reliin 4631 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | relxp 4618 | . . 3 | |
8 | 6, 7 | jctil 310 | . 2 |
9 | eleq1w 2178 | . . . . . . . 8 | |
10 | 9 | cbvexv 1872 | . . . . . . 7 |
11 | r19.28mv 3425 | . . . . . . 7 | |
12 | 10, 11 | sylbir 134 | . . . . . 6 |
13 | 12 | bicomd 140 | . . . . 5 |
14 | eliin 3788 | . . . . . . 7 | |
15 | 14 | elv 2664 | . . . . . 6 |
16 | 15 | anbi2i 452 | . . . . 5 |
17 | opelxp 4539 | . . . . . 6 | |
18 | 17 | ralbii 2418 | . . . . 5 |
19 | 13, 16, 18 | 3bitr4g 222 | . . . 4 |
20 | opelxp 4539 | . . . 4 | |
21 | vex 2663 | . . . . . 6 | |
22 | vex 2663 | . . . . . 6 | |
23 | 21, 22 | opex 4121 | . . . . 5 |
24 | eliin 3788 | . . . . 5 | |
25 | 23, 24 | ax-mp 5 | . . . 4 |
26 | 19, 20, 25 | 3bitr4g 222 | . . 3 |
27 | 26 | eqrelrdv2 4608 | . 2 |
28 | 8, 27 | mpancom 418 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 wral 2393 wrex 2394 cvv 2660 cop 3500 ciin 3784 cxp 4507 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-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-opab 3960 df-xp 4515 df-rel 4516 |
This theorem is referenced by: xpriindim 4647 |
Copyright terms: Public domain | W3C validator |