MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmmptss Structured version   Visualization version   GIF version

Theorem dmmptss 5970
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 5969 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
32ssrab3 3978 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1522  wcel 2081  Vcvv 3437  wss 3859  cmpt 5041  dom cdm 5443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-mpt 5042  df-xp 5449  df-rel 5450  df-cnv 5451  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456
This theorem is referenced by:  mptrcl  6643  fvmptss  6646  fvmptex  6648  fvmptnf  6656  elfvmptrab1  6660  mptexg  6850  mptexw  7510  dmmpossx  7620  tposssxp  7747  mptfi  8669  cnvimamptfin  8671  cantnfres  8986  mptct  9806  arwrcl  17133  cntzrcl  18198  gsumconst  18774  psrass1lem  19845  psrass1  19873  psrass23l  19876  psrcom  19877  psrass23  19878  mpfrcl  19985  psropprmul  20089  coe1mul2  20120  lmrcl  21523  1stcrestlem  21744  ptbasfi  21873  isxms2  22741  setsmstopn  22771  tngtopn  22942  rrxmval  23691  ulmss  24668  dchrrcl  25498  gsummpt2co  30495  locfinreflem  30721  sitgclg  31217  cvmsrcl  32119  snmlval  32186  gonan0  32247  bj-fvmptunsn1  34097  eldiophb  38839  elmnc  39221  itgocn  39249  submgmrcl  43531  dmmpossx2  43863
  Copyright terms: Public domain W3C validator