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Theorem dmmpti 5936
 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 5935 . 2 𝐹 Fn 𝐴
4 fndm 5904 . 2 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
53, 4ax-mp 5 1 dom 𝐹 = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643  dom cdm 5038   Fn wfn 5799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807 This theorem is referenced by:  fvmptex  6203  resfunexg  6384  brtpos2  7245  vdwlem8  15530  lubdm  16802  glbdm  16815  dprd2dlem2  18262  dprd2dlem1  18263  dprd2da  18264  ablfac1c  18293  ablfac1eu  18295  ablfaclem2  18308  ablfaclem3  18309  elocv  19831  dfac14  21231  kqtop  21358  symgtgp  21715  eltsms  21746  ressprdsds  21986  minveclem1  23003  isi1f  23247  itg1val  23256  cmvth  23558  mvth  23559  lhop2  23582  dvfsumabs  23590  dvfsumrlim2  23599  taylthlem1  23931  taylthlem2  23932  ulmdvlem1  23958  pige3  24073  relogcn  24184  atandm  24403  atanf  24407  atancn  24463  dmarea  24484  dfarea  24487  efrlim  24496  lgamgulmlem2  24556  dchrptlem2  24790  dchrptlem3  24791  dchrisum0  25009  eleenn  25576  incistruhgr  25746  vsfval  26872  ipasslem8  27076  minvecolem1  27114  xppreima2  28830  ofpreima  28848  dmsigagen  29534  measbase  29587  sseqf  29781  ballotlem7  29924  bj-dmtopon  32242  bj-inftyexpidisj  32274  bj-elccinfty  32278  bj-minftyccb  32289  fin2so  32566  poimirlem30  32609  poimir  32612  dvtan  32630  itg2addnclem2  32632  ftc1anclem6  32660  totbndbnd  32758  comptiunov2i  37017  lhe4.4ex1a  37550  dvsinax  38801  fourierdlem62  39061  fourierdlem70  39069  fourierdlem71  39070  fourierdlem80  39079  fouriersw  39124  mndpsuppss  41946  scmsuppss  41947  lincext2  42038  aacllem  42356
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