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Mirrors > Home > ILE Home > Th. List > mpoeq123 | Unicode version |
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
mpoeq123 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1528 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() | |
2 | nfra1 2508 |
. . . 4
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3 | 1, 2 | nfan 1565 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | nfv 1528 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() | |
5 | nfcv 2319 |
. . . . 5
![]() ![]() ![]() ![]() | |
6 | nfv 1528 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() | |
7 | nfra1 2508 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 6, 7 | nfan 1565 |
. . . . 5
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9 | 5, 8 | nfralxy 2515 |
. . . 4
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10 | 4, 9 | nfan 1565 |
. . 3
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11 | nfv 1528 |
. . 3
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12 | rsp 2524 |
. . . . . . 7
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13 | rsp 2524 |
. . . . . . . . . 10
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14 | eqeq2 2187 |
. . . . . . . . . 10
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15 | 13, 14 | syl6 33 |
. . . . . . . . 9
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16 | 15 | pm5.32d 450 |
. . . . . . . 8
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17 | eleq2 2241 |
. . . . . . . . 9
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18 | 17 | anbi1d 465 |
. . . . . . . 8
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19 | 16, 18 | sylan9bbr 463 |
. . . . . . 7
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20 | 12, 19 | syl6 33 |
. . . . . 6
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21 | 20 | pm5.32d 450 |
. . . . 5
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22 | eleq2 2241 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 22 | anbi1d 465 |
. . . . 5
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24 | 21, 23 | sylan9bbr 463 |
. . . 4
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25 | anass 401 |
. . . 4
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26 | anass 401 |
. . . 4
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27 | 24, 25, 26 | 3bitr4g 223 |
. . 3
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28 | 3, 10, 11, 27 | oprabbid 5922 |
. 2
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29 | df-mpo 5874 |
. 2
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30 | df-mpo 5874 |
. 2
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31 | 28, 29, 30 | 3eqtr4g 2235 |
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-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-oprab 5873 df-mpo 5874 |
This theorem is referenced by: mpoeq12 5929 mapxpen 6842 |
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