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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2257 |
. . . 4
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2 | 1 | cbvexv 1933 |
. . 3
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3 | r19.28mv 3543 |
. . . . 5
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4 | eliin 3921 |
. . . . . . 7
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5 | 4 | elv 2767 |
. . . . . 6
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6 | vex 2766 |
. . . . . . . 8
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7 | 6 | elixp 6764 |
. . . . . . 7
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8 | 7 | ralbii 2503 |
. . . . . 6
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9 | 5, 8 | bitri 184 |
. . . . 5
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10 | 6 | elixp 6764 |
. . . . . 6
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11 | vex 2766 |
. . . . . . . . . . 11
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12 | 6, 11 | fvex 5578 |
. . . . . . . . . 10
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13 | eliin 3921 |
. . . . . . . . . 10
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14 | 12, 13 | ax-mp 5 |
. . . . . . . . 9
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15 | 14 | ralbii 2503 |
. . . . . . . 8
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16 | ralcom 2660 |
. . . . . . . 8
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17 | 15, 16 | bitri 184 |
. . . . . . 7
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18 | 17 | anbi2i 457 |
. . . . . 6
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19 | 10, 18 | bitri 184 |
. . . . 5
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20 | 3, 9, 19 | 3bitr4g 223 |
. . . 4
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21 | 20 | eqrdv 2194 |
. . 3
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22 | 2, 21 | sylbir 135 |
. 2
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23 | 22 | eqcomd 2202 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ral 2480 df-rex 2481 df-v 2765 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-iin 3919 df-br 4034 df-opab 4095 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-iota 5219 df-fun 5260 df-fn 5261 df-fv 5266 df-ixp 6758 |
This theorem is referenced by: ixpintm 6784 |
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