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Theorem elmapfn 7840
Description: A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.)
Assertion
Ref Expression
elmapfn (𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴 Fn 𝐶)

Proof of Theorem elmapfn
StepHypRef Expression
1 elmapi 7839 . 2 (𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴:𝐶𝐵)
2 ffn 6012 . 2 (𝐴:𝐶𝐵𝐴 Fn 𝐶)
31, 2syl 17 1 (𝐴 ∈ (𝐵𝑚 𝐶) → 𝐴 Fn 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987   Fn wfn 5852  wf 5853  (class class class)co 6615  𝑚 cmap 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-map 7819
This theorem is referenced by:  mapxpen  8086  fsuppmapnn0fiublem  12745  fsuppmapnn0fiub  12746  fsuppmapnn0fiubOLD  12747  fsuppmapnn0fiub0  12749  suppssfz  12750  fsuppmapnn0ub  12751  frlmbas  20039  frlmsslsp  20075  eqmat  20170  matplusgcell  20179  matsubgcell  20180  matvscacell  20182  cramerlem1  20433  tmdgsum  21839  matmpt2  29693  1smat1  29694  unccur  33063  matunitlindflem1  33076  matunitlindflem2  33077  poimirlem4  33084  poimirlem5  33085  poimirlem6  33086  poimirlem7  33087  poimirlem10  33090  poimirlem11  33091  poimirlem12  33092  poimirlem16  33096  poimirlem19  33099  poimirlem29  33109  poimirlem30  33110  poimirlem31  33111  broucube  33114  rfovcnvf1od  37819  dssmapnvod  37835  dssmapntrcls  37947  k0004lem3  37968  unirnmap  38909  unirnmapsn  38915  ssmapsn  38917  dvnprodlem1  39498  dvnprodlem3  39500  rrxsnicc  39857  ioorrnopnlem  39861  ovnsubaddlem1  40121  hoiqssbllem1  40173  iccpartrn  40694  iccpartf  40695  iccpartnel  40702  mndpsuppss  41470  mndpfsupp  41475  dflinc2  41517  lincsum  41536  lincresunit2  41585
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