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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 4695 | . . . . . 6 | |
2 | 1 | rgenw 2512 | . . . . 5 |
3 | r19.2m 3480 | . . . . 5 | |
4 | 2, 3 | mpan2 422 | . . . 4 |
5 | reliin 4708 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | relxp 4695 | . . 3 | |
8 | 6, 7 | jctil 310 | . 2 |
9 | eleq1w 2218 | . . . . . . . 8 | |
10 | 9 | cbvexv 1898 | . . . . . . 7 |
11 | r19.28mv 3486 | . . . . . . 7 | |
12 | 10, 11 | sylbir 134 | . . . . . 6 |
13 | 12 | bicomd 140 | . . . . 5 |
14 | eliin 3854 | . . . . . . 7 | |
15 | 14 | elv 2716 | . . . . . 6 |
16 | 15 | anbi2i 453 | . . . . 5 |
17 | opelxp 4616 | . . . . . 6 | |
18 | 17 | ralbii 2463 | . . . . 5 |
19 | 13, 16, 18 | 3bitr4g 222 | . . . 4 |
20 | opelxp 4616 | . . . 4 | |
21 | vex 2715 | . . . . . 6 | |
22 | vex 2715 | . . . . . 6 | |
23 | 21, 22 | opex 4189 | . . . . 5 |
24 | eliin 3854 | . . . . 5 | |
25 | 23, 24 | ax-mp 5 | . . . 4 |
26 | 19, 20, 25 | 3bitr4g 222 | . . 3 |
27 | 26 | eqrelrdv2 4685 | . 2 |
28 | 8, 27 | mpancom 419 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 wral 2435 wrex 2436 cvv 2712 cop 3563 ciin 3850 cxp 4584 wrel 4591 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-iin 3852 df-opab 4026 df-xp 4592 df-rel 4593 |
This theorem is referenced by: xpriindim 4724 |
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