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Theorem dmmptss 6198
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 6197 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
32ssrab3 4045 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3448  wss 3915  cmpt 5193  dom cdm 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-mpt 5194  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651
This theorem is referenced by:  mptrcl  6962  fvmptss  6965  fvmptex  6967  fvmptnf  6975  elfvmptrab1w  6979  elfvmptrab1  6980  mptexg  7176  mptexw  7890  dmmpossx  8003  tposssxp  8166  mptfi  9302  cnvimamptfin  9304  cantnfres  9620  mptct  10481  arwrcl  17937  cntzrcl  19114  gsumconst  19718  psrass1lemOLD  21358  psrass1lem  21361  psrass1  21390  psrass23l  21393  psrcom  21394  psrass23  21395  mpfrcl  21511  psropprmul  21625  coe1mul2  21656  lmrcl  22598  1stcrestlem  22819  ptbasfi  22948  isxms2  23817  setsmstopn  23849  tngtopn  24030  rrxmval  24785  ulmss  25772  dchrrcl  26604  gsummpt2co  31932  locfinreflem  32461  sitgclg  32982  cvmsrcl  33898  snmlval  33965  gonan0  34026  bj-fvmptunsn1  35757  eldiophb  41109  elmnc  41492  itgocn  41520  submgmrcl  46150  dmmpossx2  46486
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