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Mirrors > Home > MPE Home > Th. List > elmapfn | Structured version Visualization version GIF version |
Description: A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.) |
Ref | Expression |
---|---|
elmapfn | ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴 Fn 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8839 | . 2 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴:𝐶⟶𝐵) | |
2 | 1 | ffnd 6715 | 1 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴 Fn 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Fn wfn 6535 (class class class)co 7404 ↑m cmap 8816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-map 8818 |
This theorem is referenced by: mapxpen 9139 fsuppmapnn0fiublem 13951 fsuppmapnn0fiub 13952 fsuppmapnn0fiub0 13954 suppssfz 13955 fsuppmapnn0ub 13956 frlmbas 21294 frlmsslsp 21335 eqmat 21908 matplusgcell 21917 matsubgcell 21918 matvscacell 21920 cramerlem1 22171 tmdgsum 23581 fmptco1f1o 31835 islinds5 32449 ellspds 32450 lbsdiflsp0 32656 matmpo 32721 1smat1 32722 actfunsnf1o 33554 actfunsnrndisj 33555 reprinfz1 33572 unccur 36409 matunitlindflem1 36422 matunitlindflem2 36423 poimirlem4 36430 poimirlem5 36431 poimirlem6 36432 poimirlem7 36433 poimirlem10 36436 poimirlem11 36437 poimirlem12 36438 poimirlem16 36442 poimirlem19 36445 poimirlem29 36455 poimirlem30 36456 poimirlem31 36457 broucube 36460 fsuppind 41112 ofoafo 42039 ofoaass 42043 ofoacom 42044 rfovcnvf1od 42688 dssmapnvod 42704 dssmapntrcls 42812 k0004lem3 42833 unirnmap 43840 unirnmapsn 43846 ssmapsn 43848 dvnprodlem1 44597 dvnprodlem3 44599 rrxsnicc 44951 ioorrnopnlem 44955 ovnsubaddlem1 45221 hoiqssbllem1 45273 iccpartrn 46033 iccpartf 46034 iccpartnel 46041 mndpsuppss 46949 mndpfsupp 46954 dflinc2 46993 lincsum 47012 lincresunit2 47061 2arymaptfo 47242 rrx2pnecoorneor 47303 rrx2linest 47330 |
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