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| Mirrors > Home > MPE Home > Th. List > funmpt | Structured version Visualization version GIF version | ||
| Description: A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
| Ref | Expression |
|---|---|
| funmpt | ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funopab4 6603 | . 2 ⊢ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
| 2 | df-mpt 5226 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
| 3 | 2 | funeqi 6587 | . 2 ⊢ (Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ Fun {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}) |
| 4 | 1, 3 | mpbir 231 | 1 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {copab 5205 ↦ cmpt 5225 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 |
| This theorem is referenced by: funmpt2 6605 resfunexg 7235 mptexg 7241 mptexgf 7242 mptexw 7977 brtpos2 8257 tposfun 8267 mptfi 9391 fsuppssov1 9424 sniffsupp 9440 cantnfrescl 9716 cantnflem1 9729 r0weon 10052 axcc2lem 10476 mptct 10578 negfi 12217 mptnn0fsupp 14038 ccatalpha 14631 mreacs 17701 acsfn 17702 isofval 17801 lubfun 18397 glbfun 18410 acsficl2d 18597 gsum2dlem2 19989 gsum2d 19990 dprdfinv 20039 dprdfadd 20040 dmdprdsplitlem 20057 dpjidcl 20078 mptscmfsupp0 20925 pjpm 21728 frlmphllem 21800 uvcff 21811 uvcresum 21813 psrass1lem 21952 psrlidm 21982 psrridm 21983 psrass1 21984 psrass23l 21987 psrcom 21988 psrass23 21989 mplsubrg 22025 mplmon 22053 mplmonmul 22054 mplcoe1 22055 mplcoe5 22058 mplbas2 22060 evlslem2 22103 evlslem6 22105 psdmplcl 22166 psdmul 22170 psropprmul 22239 coe1mul2 22272 evls1fpws 22373 oftpos 22458 pmatcollpw2lem 22783 tgrest 23167 cmpfi 23416 1stcrestlem 23460 ptcnplem 23629 xkoinjcn 23695 symgtgp 24114 eltsms 24141 rrxmval 25439 tdeglem4 26099 plypf1 26251 tayl0 26403 taylthlem1 26415 xrlimcnp 27011 nosupno 27748 noinfno 27763 abrexexd 32528 ofpreima 32675 fisuppov1 32692 mptiffisupp 32702 mptctf 32729 gsummptres2 33056 psgnfzto1stlem 33120 rmfsupp2 33242 elrspunidl 33456 elrspunsn 33457 locfinreflem 33839 measdivcstALTV 34226 sitgf 34349 imageval 35931 poimirlem30 37657 poimir 37660 evlsvvvallem2 42572 evlsvvval 42573 selvvvval 42595 evlselv 42597 mhphf 42607 choicefi 45205 rn1st 45280 fourierdlem80 46201 sge0tsms 46395 scmsuppss 48287 rmfsupp 48289 scmfsupp 48291 fdivval 48460 |
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