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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 2652 |
. 2
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2 | nfcv 2240 |
. . . . . 6
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3 | nfcsb1v 2985 |
. . . . . 6
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4 | csbeq1a 2963 |
. . . . . 6
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5 | 2, 3, 4 | cbvixp 6539 |
. . . . 5
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6 | 5 | eleq2i 2166 |
. . . 4
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7 | elixp2 6526 |
. . . 4
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8 | 3anass 934 |
. . . 4
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9 | 6, 7, 8 | 3bitri 205 |
. . 3
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10 | eqid 2100 |
. . . . . . . 8
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11 | 10 | fnmpt 5185 |
. . . . . . 7
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12 | 10 | fvmpt2 5436 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | simpr 109 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 12, 13 | eqeltrd 2176 |
. . . . . . . 8
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15 | 14 | ralimiaa 2453 |
. . . . . . 7
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16 | 11, 15 | jca 302 |
. . . . . 6
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17 | dffn2 5210 |
. . . . . . . 8
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18 | 10 | fmpt 5502 |
. . . . . . . . 9
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19 | 10 | fvmpt2 5436 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 19 | eleq1d 2168 |
. . . . . . . . . . . 12
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21 | 20 | biimpd 143 |
. . . . . . . . . . 11
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22 | 21 | ralimiaa 2453 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | ralim 2450 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | syl 14 |
. . . . . . . . 9
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25 | 18, 24 | sylbir 134 |
. . . . . . . 8
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26 | 17, 25 | sylbi 120 |
. . . . . . 7
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27 | 26 | imp 123 |
. . . . . 6
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28 | 16, 27 | impbii 125 |
. . . . 5
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29 | nfv 1476 |
. . . . . . 7
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30 | nffvmpt1 5364 |
. . . . . . . 8
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31 | 30, 3 | nfel 2249 |
. . . . . . 7
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32 | fveq2 5353 |
. . . . . . . 8
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33 | 32, 4 | eleq12d 2170 |
. . . . . . 7
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34 | 29, 31, 33 | cbvral 2608 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34 | anbi2i 448 |
. . . . 5
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36 | 28, 35 | bitri 183 |
. . . 4
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37 | mptexg 5577 |
. . . . 5
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38 | 37 | biantrurd 301 |
. . . 4
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39 | 36, 38 | syl5rbb 192 |
. . 3
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40 | 9, 39 | syl5bb 191 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 1, 40 | syl 14 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ixp 6523 |
This theorem is referenced by: (None) |
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