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Mirrors > Home > ILE Home > Th. List > xpiundir | Unicode version |
Description: Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.) |
Ref | Expression |
---|---|
xpiundir |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 2753 | . . . . 5 | |
2 | df-rex 2454 | . . . . . 6 | |
3 | 2 | rexbii 2477 | . . . . 5 |
4 | eliun 3875 | . . . . . . . 8 | |
5 | 4 | anbi1i 455 | . . . . . . 7 |
6 | r19.41v 2626 | . . . . . . 7 | |
7 | 5, 6 | bitr4i 186 | . . . . . 6 |
8 | 7 | exbii 1598 | . . . . 5 |
9 | 1, 3, 8 | 3bitr4ri 212 | . . . 4 |
10 | df-rex 2454 | . . . 4 | |
11 | elxp2 4627 | . . . . 5 | |
12 | 11 | rexbii 2477 | . . . 4 |
13 | 9, 10, 12 | 3bitr4i 211 | . . 3 |
14 | elxp2 4627 | . . 3 | |
15 | eliun 3875 | . . 3 | |
16 | 13, 14, 15 | 3bitr4i 211 | . 2 |
17 | 16 | eqriv 2167 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1348 wex 1485 wcel 2141 wrex 2449 cop 3584 ciun 3871 cxp 4607 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-iun 3873 df-opab 4049 df-xp 4615 |
This theorem is referenced by: iunxpconst 4669 resiun2 4909 txbasval 12982 |
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