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Mirrors > Home > ILE Home > Th. List > mpoxopovel | Unicode version |
Description: Element of the 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 and Mario Carneiro, 10-Oct-2017.) |
Ref | Expression |
---|---|
mpoxopoveq.f |
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Ref | Expression |
---|---|
mpoxopovel |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoxopoveq.f |
. . . 4
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2 | 1 | mpoxopn0yelv 6234 |
. . 3
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3 | 2 | pm4.71rd 394 |
. 2
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4 | 1 | mpoxopoveq 6235 |
. . . . . 6
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5 | 4 | eleq2d 2247 |
. . . . 5
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6 | nfcv 2319 |
. . . . . . 7
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7 | 6 | elrabsf 3001 |
. . . . . 6
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8 | sbccom 3038 |
. . . . . . . 8
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9 | sbccom 3038 |
. . . . . . . . 9
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10 | 9 | sbcbii 3022 |
. . . . . . . 8
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11 | 8, 10 | bitri 184 |
. . . . . . 7
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12 | 11 | anbi2i 457 |
. . . . . 6
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13 | 7, 12 | bitri 184 |
. . . . 5
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14 | 5, 13 | bitrdi 196 |
. . . 4
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15 | 14 | pm5.32da 452 |
. . 3
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16 | 3anass 982 |
. . 3
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17 | 15, 16 | bitr4di 198 |
. 2
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18 | 3, 17 | bitrd 188 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 |
This theorem is referenced by: (None) |
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