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Mirrors > Home > ILE Home > Th. List > mptelixpg | Unicode version |
Description: Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.) |
Ref | Expression |
---|---|
mptelixpg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | nfcv 2332 |
. . . . . 6
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3 | nfcsb1v 3105 |
. . . . . 6
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4 | csbeq1a 3081 |
. . . . . 6
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5 | 2, 3, 4 | cbvixp 6740 |
. . . . 5
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6 | 5 | eleq2i 2256 |
. . . 4
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7 | elixp2 6727 |
. . . 4
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8 | 3anass 984 |
. . . 4
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9 | 6, 7, 8 | 3bitri 206 |
. . 3
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10 | eqid 2189 |
. . . . . . . 8
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11 | 10 | fnmpt 5361 |
. . . . . . 7
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12 | 10 | fvmpt2 5619 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | simpr 110 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 12, 13 | eqeltrd 2266 |
. . . . . . . 8
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15 | 14 | ralimiaa 2552 |
. . . . . . 7
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16 | 11, 15 | jca 306 |
. . . . . 6
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17 | dffn2 5386 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 10 | fmpt 5686 |
. . . . . . . . 9
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19 | 10 | fvmpt2 5619 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 19 | eleq1d 2258 |
. . . . . . . . . . . 12
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21 | 20 | biimpd 144 |
. . . . . . . . . . 11
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22 | 21 | ralimiaa 2552 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | ralim 2549 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | syl 14 |
. . . . . . . . 9
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25 | 18, 24 | sylbir 135 |
. . . . . . . 8
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26 | 17, 25 | sylbi 121 |
. . . . . . 7
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27 | 26 | imp 124 |
. . . . . 6
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28 | 16, 27 | impbii 126 |
. . . . 5
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29 | nfv 1539 |
. . . . . . 7
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30 | nffvmpt1 5545 |
. . . . . . . 8
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31 | 30, 3 | nfel 2341 |
. . . . . . 7
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32 | fveq2 5534 |
. . . . . . . 8
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33 | 32, 4 | eleq12d 2260 |
. . . . . . 7
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34 | 29, 31, 33 | cbvral 2714 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34 | anbi2i 457 |
. . . . 5
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36 | 28, 35 | bitri 184 |
. . . 4
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37 | mptexg 5761 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
38 | 37 | biantrurd 305 |
. . . 4
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39 | 36, 38 | bitr2id 193 |
. . 3
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40 | 9, 39 | bitrid 192 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 1, 40 | syl 14 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ixp 6724 |
This theorem is referenced by: (None) |
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