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Mirrors > Home > ILE Home > Th. List > ixpiinm | Unicode version |
Description: The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.) |
Ref | Expression |
---|---|
ixpiinm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2200 | . . . 4 | |
2 | 1 | cbvexv 1890 | . . 3 |
3 | r19.28mv 3455 | . . . . 5 | |
4 | eliin 3818 | . . . . . . 7 | |
5 | 4 | elv 2690 | . . . . . 6 |
6 | vex 2689 | . . . . . . . 8 | |
7 | 6 | elixp 6599 | . . . . . . 7 |
8 | 7 | ralbii 2441 | . . . . . 6 |
9 | 5, 8 | bitri 183 | . . . . 5 |
10 | 6 | elixp 6599 | . . . . . 6 |
11 | vex 2689 | . . . . . . . . . . 11 | |
12 | 6, 11 | fvex 5441 | . . . . . . . . . 10 |
13 | eliin 3818 | . . . . . . . . . 10 | |
14 | 12, 13 | ax-mp 5 | . . . . . . . . 9 |
15 | 14 | ralbii 2441 | . . . . . . . 8 |
16 | ralcom 2594 | . . . . . . . 8 | |
17 | 15, 16 | bitri 183 | . . . . . . 7 |
18 | 17 | anbi2i 452 | . . . . . 6 |
19 | 10, 18 | bitri 183 | . . . . 5 |
20 | 3, 9, 19 | 3bitr4g 222 | . . . 4 |
21 | 20 | eqrdv 2137 | . . 3 |
22 | 2, 21 | sylbir 134 | . 2 |
23 | 22 | eqcomd 2145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wral 2416 cvv 2686 ciin 3814 wfn 5118 cfv 5123 cixp 6592 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iin 3816 df-br 3930 df-opab 3990 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-ixp 6593 |
This theorem is referenced by: ixpintm 6619 |
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