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 4720 | . . . . . 6 | |
2 | 1 | rgenw 2525 | . . . . 5 |
3 | r19.2m 3501 | . . . . 5 | |
4 | 2, 3 | mpan2 423 | . . . 4 |
5 | reliin 4733 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | relxp 4720 | . . 3 | |
8 | 6, 7 | jctil 310 | . 2 |
9 | eleq1w 2231 | . . . . . . . 8 | |
10 | 9 | cbvexv 1911 | . . . . . . 7 |
11 | r19.28mv 3507 | . . . . . . 7 | |
12 | 10, 11 | sylbir 134 | . . . . . 6 |
13 | 12 | bicomd 140 | . . . . 5 |
14 | eliin 3878 | . . . . . . 7 | |
15 | 14 | elv 2734 | . . . . . 6 |
16 | 15 | anbi2i 454 | . . . . 5 |
17 | opelxp 4641 | . . . . . 6 | |
18 | 17 | ralbii 2476 | . . . . 5 |
19 | 13, 16, 18 | 3bitr4g 222 | . . . 4 |
20 | opelxp 4641 | . . . 4 | |
21 | vex 2733 | . . . . . 6 | |
22 | vex 2733 | . . . . . 6 | |
23 | 21, 22 | opex 4214 | . . . . 5 |
24 | eliin 3878 | . . . . 5 | |
25 | 23, 24 | ax-mp 5 | . . . 4 |
26 | 19, 20, 25 | 3bitr4g 222 | . . 3 |
27 | 26 | eqrelrdv2 4710 | . 2 |
28 | 8, 27 | mpancom 420 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 wral 2448 wrex 2449 cvv 2730 cop 3586 ciin 3874 cxp 4609 wrel 4616 |
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 4107 ax-pow 4160 ax-pr 4194 |
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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-iin 3876 df-opab 4051 df-xp 4617 df-rel 4618 |
This theorem is referenced by: xpriindim 4749 |
Copyright terms: Public domain | W3C validator |