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Theorem dmmpti 5990
Description: Domain of the mapping operation. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fnmpti.1 𝐵 ∈ V
fnmpti.2 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
dmmpti dom 𝐹 = 𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem dmmpti
StepHypRef Expression
1 fnmpti.1 . . 3 𝐵 ∈ V
2 fnmpti.2 . . 3 𝐹 = (𝑥𝐴𝐵)
31, 2fnmpti 5989 . 2 𝐹 Fn 𝐴
4 fndm 5958 . 2 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
53, 4ax-mp 5 1 dom 𝐹 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3190  cmpt 4683  dom cdm 5084   Fn wfn 5852
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
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-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  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-fun 5859  df-fn 5860
This theorem is referenced by:  fvmptex  6261  resfunexg  6444  brtpos2  7318  vdwlem8  15635  lubdm  16919  glbdm  16932  dprd2dlem2  18379  dprd2dlem1  18380  dprd2da  18381  ablfac1c  18410  ablfac1eu  18412  ablfaclem2  18425  ablfaclem3  18426  elocv  19952  dmtopon  20667  dfac14  21361  kqtop  21488  symgtgp  21845  eltsms  21876  ressprdsds  22116  minveclem1  23135  isi1f  23381  itg1val  23390  cmvth  23692  mvth  23693  lhop2  23716  dvfsumabs  23724  dvfsumrlim2  23733  taylthlem1  24065  taylthlem2  24066  ulmdvlem1  24092  pige3  24207  relogcn  24318  atandm  24537  atanf  24541  atancn  24597  dmarea  24618  dfarea  24621  efrlim  24630  lgamgulmlem2  24690  dchrptlem2  24924  dchrptlem3  24925  dchrisum0  25143  eleenn  25710  incistruhgr  25904  vsfval  27376  ipasslem8  27580  minvecolem1  27618  xppreima2  29333  ofpreima  29349  dmsigagen  30030  measbase  30083  sseqf  30277  ballotlem7  30420  nosino  31628  nosires  31630  bj-inftyexpidisj  32769  bj-elccinfty  32773  bj-minftyccb  32784  fin2so  33067  poimirlem30  33110  poimir  33113  dvtan  33131  itg2addnclem2  33133  ftc1anclem6  33161  totbndbnd  33259  comptiunov2i  37518  lhe4.4ex1a  38049  dvsinax  39463  fourierdlem62  39722  fourierdlem70  39730  fourierdlem71  39731  fourierdlem80  39740  fouriersw  39785  smflimsuplem1  40363  smflimsuplem4  40366  mndpsuppss  41470  scmsuppss  41471  lincext2  41562  aacllem  41880
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