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Theorem dmmptss 6252
Description: The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
Hypothesis
Ref Expression
dmmpt.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
dmmptss dom 𝐹𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem dmmptss
StepHypRef Expression
1 dmmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
21dmmpt 6251 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
32ssrab3 4079 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  Vcvv 3462  wss 3947  cmpt 5236  dom cdm 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-mpt 5237  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695
This theorem is referenced by:  mptrcl  7018  fvmptss  7021  fvmptex  7023  fvmptnf  7031  elfvmptrab1w  7036  elfvmptrab1  7037  mptexg  7238  mptexw  7966  dmmpossx  8080  tposssxp  8245  mptfi  9395  cnvimamptfin  9397  cantnfres  9720  mptct  10581  arwrcl  18066  submgmrcl  18688  cntzrcl  19321  gsumconst  19932  psrass1lemOLD  21938  psrass1lem  21941  psrass1  21973  psrass23l  21976  psrcom  21977  psrass23  21978  mpfrcl  22100  psropprmul  22227  coe1mul2  22260  lmrcl  23226  1stcrestlem  23447  ptbasfi  23576  isxms2  24445  setsmstopn  24477  tngtopn  24658  rrxmval  25424  ulmss  26426  dchrrcl  27269  gsummpt2co  32916  locfinreflem  33655  sitgclg  34176  cvmsrcl  35092  snmlval  35159  gonan0  35220  bj-fvmptunsn1  36964  eldiophb  42414  elmnc  42797  itgocn  42825  dmmpossx2  47715
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