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Mirrors > Home > MPE Home > Th. List > reldv | Structured version Unicode version |
Description: The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
reldv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5042 |
. . . . . . . 8
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2 | 1 | rgenw 2888 |
. . . . . . 7
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3 | reliun 5055 |
. . . . . . 7
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4 | 2, 3 | mpbir 209 |
. . . . . 6
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5 | df-rel 4942 |
. . . . . 6
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6 | 4, 5 | mpbi 208 |
. . . . 5
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7 | 6 | rgenw 2888 |
. . . 4
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8 | 7 | rgenw 2888 |
. . 3
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9 | df-dv 21455 |
. . . 4
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10 | 9 | ovmptss 6751 |
. . 3
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11 | 8, 10 | ax-mp 5 |
. 2
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12 | df-rel 4942 |
. 2
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13 | 11, 12 | mpbir 209 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4508 ax-nul 4516 ax-pow 4565 ax-pr 4626 ax-un 6469 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2599 df-ne 2644 df-ral 2798 df-rex 2799 df-rab 2802 df-v 3067 df-sbc 3282 df-csb 3384 df-dif 3426 df-un 3428 df-in 3430 df-ss 3437 df-nul 3733 df-if 3887 df-sn 3973 df-pr 3975 df-op 3979 df-uni 4187 df-iun 4268 df-br 4388 df-opab 4446 df-mpt 4447 df-id 4731 df-xp 4941 df-rel 4942 df-cnv 4943 df-co 4944 df-dm 4945 df-rn 4946 df-res 4947 df-ima 4948 df-iota 5476 df-fun 5515 df-fv 5521 df-ov 6190 df-oprab 6191 df-mpt2 6192 df-1st 6674 df-2nd 6675 df-dv 21455 |
This theorem is referenced by: perfdvf 21491 dvres 21499 dvres3 21501 dvres3a 21502 dvidlem 21503 dvmulbr 21526 dvaddf 21529 dvmulf 21530 dvcobr 21533 dvcof 21535 dvcnv 21562 |
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