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Mirrors > Home > ILE Home > Th. List > eloprabi | Unicode version |
Description: A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
eloprabi.1 |
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eloprabi.2 |
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eloprabi.3 |
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Ref | Expression |
---|---|
eloprabi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2043 |
. . . . . 6
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2 | 1 | anbi1d 438 |
. . . . 5
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3 | 2 | 3exbidv 1746 |
. . . 4
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4 | df-oprab 5459 |
. . . 4
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5 | 3, 4 | elab2g 2683 |
. . 3
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6 | 5 | ibi 165 |
. 2
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7 | vex 2554 |
. . . . . . . . . . . 12
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8 | vex 2554 |
. . . . . . . . . . . 12
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9 | 7, 8 | opex 3957 |
. . . . . . . . . . 11
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10 | vex 2554 |
. . . . . . . . . . 11
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11 | 9, 10 | op1std 5717 |
. . . . . . . . . 10
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12 | 11 | fveq2d 5125 |
. . . . . . . . 9
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13 | 7, 8 | op1st 5715 |
. . . . . . . . 9
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14 | 12, 13 | syl6req 2086 |
. . . . . . . 8
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15 | eloprabi.1 |
. . . . . . . 8
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16 | 14, 15 | syl 14 |
. . . . . . 7
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17 | 11 | fveq2d 5125 |
. . . . . . . . 9
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18 | 7, 8 | op2nd 5716 |
. . . . . . . . 9
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19 | 17, 18 | syl6req 2086 |
. . . . . . . 8
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20 | eloprabi.2 |
. . . . . . . 8
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21 | 19, 20 | syl 14 |
. . . . . . 7
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22 | 9, 10 | op2ndd 5718 |
. . . . . . . . 9
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23 | 22 | eqcomd 2042 |
. . . . . . . 8
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24 | eloprabi.3 |
. . . . . . . 8
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25 | 23, 24 | syl 14 |
. . . . . . 7
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26 | 16, 21, 25 | 3bitrd 203 |
. . . . . 6
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27 | 26 | biimpa 280 |
. . . . 5
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28 | 27 | exlimiv 1486 |
. . . 4
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29 | 28 | exlimiv 1486 |
. . 3
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30 | 29 | exlimiv 1486 |
. 2
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31 | 6, 30 | syl 14 |
1
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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 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fv 4853 df-oprab 5459 df-1st 5709 df-2nd 5710 |
This theorem is referenced by: (None) |
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