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Mirrors > Home > ILE Home > Th. List > mpoxopoveq | Unicode version |
Description: Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.) |
Ref | Expression |
---|---|
mpoxopoveq.f |
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Ref | Expression |
---|---|
mpoxopoveq |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoxopoveq.f |
. . 3
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2 | 1 | a1i 9 |
. 2
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3 | fveq2 5554 |
. . . . 5
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4 | op1stg 6203 |
. . . . . 6
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5 | 4 | adantr 276 |
. . . . 5
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6 | 3, 5 | sylan9eqr 2248 |
. . . 4
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7 | 6 | adantrr 479 |
. . 3
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8 | sbceq1a 2995 |
. . . . . 6
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9 | 8 | adantl 277 |
. . . . 5
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10 | 9 | adantl 277 |
. . . 4
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11 | sbceq1a 2995 |
. . . . . 6
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12 | 11 | adantr 276 |
. . . . 5
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13 | 12 | adantl 277 |
. . . 4
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14 | 10, 13 | bitrd 188 |
. . 3
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15 | 7, 14 | rabeqbidv 2755 |
. 2
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16 | opexg 4257 |
. . 3
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17 | 16 | adantr 276 |
. 2
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18 | simpr 110 |
. 2
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19 | rabexg 4172 |
. . 3
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20 | 19 | ad2antrr 488 |
. 2
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21 | equid 1712 |
. . 3
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22 | nfvd 1540 |
. . 3
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23 | 21, 22 | ax-mp 5 |
. 2
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24 | nfvd 1540 |
. . 3
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25 | 21, 24 | ax-mp 5 |
. 2
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26 | nfcv 2336 |
. 2
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27 | nfcv 2336 |
. 2
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28 | nfsbc1v 3004 |
. . 3
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29 | nfcv 2336 |
. . 3
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30 | 28, 29 | nfrabw 2675 |
. 2
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31 | nfsbc1v 3004 |
. . . 4
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32 | 26, 31 | nfsbc 3006 |
. . 3
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33 | nfcv 2336 |
. . 3
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34 | 32, 33 | nfrabw 2675 |
. 2
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35 | 2, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34 | ovmpodxf 6044 |
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-in1 615 ax-in2 616 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-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-iota 5215 df-fun 5256 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 |
This theorem is referenced by: mpoxopovel 6294 |
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