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Theorem dmmptss 6088
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 6087 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
32ssrab3 4054 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  cmpt 5137  dom cdm 5548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  mptrcl  6769  fvmptss  6772  fvmptex  6774  fvmptnf  6782  elfvmptrab1w  6786  elfvmptrab1  6787  mptexg  6975  mptexw  7643  dmmpossx  7753  tposssxp  7885  mptfi  8811  cnvimamptfin  8813  cantnfres  9128  mptct  9948  arwrcl  17292  cntzrcl  18395  gsumconst  18983  psrass1lem  20085  psrass1  20113  psrass23l  20116  psrcom  20117  psrass23  20118  mpfrcl  20226  psropprmul  20334  coe1mul2  20365  lmrcl  21767  1stcrestlem  21988  ptbasfi  22117  isxms2  22985  setsmstopn  23015  tngtopn  23186  rrxmval  23935  ulmss  24912  dchrrcl  25743  gsummpt2co  30613  locfinreflem  31003  sitgclg  31499  cvmsrcl  32408  snmlval  32475  gonan0  32536  bj-fvmptunsn1  34431  eldiophb  39232  elmnc  39614  itgocn  39642  submgmrcl  43926  dmmpossx2  44313
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