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Theorem dmmptss 6261
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 6260 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
32ssrab3 4082 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  cmpt 5225  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  mptrcl  7025  fvmptss  7028  fvmptex  7030  fvmptnf  7038  elfvmptrab1w  7043  elfvmptrab1  7044  mptexg  7241  mptexw  7977  dmmpossx  8091  tposssxp  8255  mptfi  9391  cnvimamptfin  9393  cantnfres  9717  mptct  10578  arwrcl  18089  submgmrcl  18708  cntzrcl  19345  gsumconst  19952  psrass1lem  21952  psrass1  21984  psrass23l  21987  psrcom  21988  psrass23  21989  mpfrcl  22109  psropprmul  22239  coe1mul2  22272  lmrcl  23239  1stcrestlem  23460  ptbasfi  23589  isxms2  24458  setsmstopn  24490  tngtopn  24671  rrxmval  25439  ulmss  26440  dchrrcl  27284  gsummpt2co  33051  locfinreflem  33839  sitgclg  34344  cvmsrcl  35269  snmlval  35336  gonan0  35397  bj-fvmptunsn1  37258  eldiophb  42768  elmnc  43148  itgocn  43176  dmmpossx2  48253  dmtposss  48776
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