Theorem dmmptss 6272

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 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
dmmptss	⊢ dom 𝐹 ⊆ 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem dmmptss

Step	Hyp	Ref	Expression
1		dmmpt.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2	1	dmmpt 6271	. 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
3	2	ssrab3 4105	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976   ↦ cmpt 5249  dom cdm 5700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  mptrcl  7038  fvmptss  7041  fvmptex  7043  fvmptnf  7051  elfvmptrab1w  7056  elfvmptrab1  7057  mptexg  7258  mptexw  7993  dmmpossx  8107  tposssxp  8271  mptfi  9421  cnvimamptfin  9423  cantnfres  9746  mptct  10607  arwrcl  18111  submgmrcl  18733  cntzrcl  19367  gsumconst  19976  psrass1lem  21975  psrass1  22007  psrass23l  22010  psrcom  22011  psrass23  22012  mpfrcl  22132  psropprmul  22260  coe1mul2  22293  lmrcl  23260  1stcrestlem  23481  ptbasfi  23610  isxms2  24479  setsmstopn  24511  tngtopn  24692  rrxmval  25458  ulmss  26458  dchrrcl  27302  gsummpt2co  33031  locfinreflem  33786  sitgclg  34307  cvmsrcl  35232  snmlval  35299  gonan0  35360  bj-fvmptunsn1  37223  eldiophb  42713  elmnc  43093  itgocn  43121  dmmpossx2  48061