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Mirrors > Home > MPE Home > Th. List > funcf1 | Structured version Visualization version Unicode version |
Description: The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
funcf1.b |
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funcf1.c |
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funcf1.f |
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Ref | Expression |
---|---|
funcf1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcf1.f |
. . 3
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2 | funcf1.b |
. . . 4
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3 | funcf1.c |
. . . 4
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4 | eqid 2453 |
. . . 4
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5 | eqid 2453 |
. . . 4
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6 | eqid 2453 |
. . . 4
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7 | eqid 2453 |
. . . 4
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8 | eqid 2453 |
. . . 4
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9 | eqid 2453 |
. . . 4
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10 | df-br 4406 |
. . . . . . 7
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11 | 1, 10 | sylib 200 |
. . . . . 6
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12 | funcrcl 15780 |
. . . . . 6
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13 | 11, 12 | syl 17 |
. . . . 5
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14 | 13 | simpld 461 |
. . . 4
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15 | 13 | simprd 465 |
. . . 4
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16 | 2, 3, 4, 5, 6, 7, 8, 9, 14, 15 | isfunc 15781 |
. . 3
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17 | 1, 16 | mpbid 214 |
. 2
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18 | 17 | simp1d 1021 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-map 7479 df-ixp 7528 df-func 15775 |
This theorem is referenced by: funcsect 15789 funcinv 15790 funciso 15791 funcoppc 15792 cofu1 15801 cofucl 15805 cofuass 15806 cofulid 15807 cofurid 15808 funcres 15813 funcres2 15815 wunfunc 15816 funcres2c 15818 fullpropd 15837 fthsect 15842 fthinv 15843 fthmon 15844 ffthiso 15846 cofull 15851 cofth 15852 fuccocl 15881 fucidcl 15882 fuclid 15883 fucrid 15884 fucass 15885 fucsect 15889 fucinv 15890 invfuc 15891 fuciso 15892 natpropd 15893 fucpropd 15894 catciso 16014 prfval 16096 prfcl 16100 prf1st 16101 prf2nd 16102 1st2ndprf 16103 evlfcllem 16118 evlfcl 16119 curf1cl 16125 curfcl 16129 uncf1 16133 uncf2 16134 curfuncf 16135 uncfcurf 16136 diag1cl 16139 curf2ndf 16144 yon1cl 16160 oyon1cl 16168 yonedalem3a 16171 yonedalem4c 16174 yonedalem3b 16176 yonedalem3 16177 yonedainv 16178 yonffthlem 16179 yoniso 16182 |
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