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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 4643 | . . . . . 6 | |
2 | 1 | rgenw 2485 | . . . . 5 |
3 | r19.2m 3444 | . . . . 5 | |
4 | 2, 3 | mpan2 421 | . . . 4 |
5 | reliin 4656 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | relxp 4643 | . . 3 | |
8 | 6, 7 | jctil 310 | . 2 |
9 | eleq1w 2198 | . . . . . . . 8 | |
10 | 9 | cbvexv 1890 | . . . . . . 7 |
11 | r19.28mv 3450 | . . . . . . 7 | |
12 | 10, 11 | sylbir 134 | . . . . . 6 |
13 | 12 | bicomd 140 | . . . . 5 |
14 | eliin 3813 | . . . . . . 7 | |
15 | 14 | elv 2685 | . . . . . 6 |
16 | 15 | anbi2i 452 | . . . . 5 |
17 | opelxp 4564 | . . . . . 6 | |
18 | 17 | ralbii 2439 | . . . . 5 |
19 | 13, 16, 18 | 3bitr4g 222 | . . . 4 |
20 | opelxp 4564 | . . . 4 | |
21 | vex 2684 | . . . . . 6 | |
22 | vex 2684 | . . . . . 6 | |
23 | 21, 22 | opex 4146 | . . . . 5 |
24 | eliin 3813 | . . . . 5 | |
25 | 23, 24 | ax-mp 5 | . . . 4 |
26 | 19, 20, 25 | 3bitr4g 222 | . . 3 |
27 | 26 | eqrelrdv2 4633 | . 2 |
28 | 8, 27 | mpancom 418 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wral 2414 wrex 2415 cvv 2681 cop 3525 ciin 3809 cxp 4532 wrel 4539 |
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 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-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-iin 3811 df-opab 3985 df-xp 4540 df-rel 4541 |
This theorem is referenced by: xpriindim 4672 |
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