Theorem dmmptss 6263

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 6262	. 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
3	2	ssrab3 4092	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963   ↦ cmpt 5231  dom cdm 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702

This theorem is referenced by:  mptrcl  7025  fvmptss  7028  fvmptex  7030  fvmptnf  7038  elfvmptrab1w  7043  elfvmptrab1  7044  mptexg  7241  mptexw  7976  dmmpossx  8090  tposssxp  8254  mptfi  9389  cnvimamptfin  9391  cantnfres  9715  mptct  10576  arwrcl  18098  submgmrcl  18721  cntzrcl  19358  gsumconst  19967  psrass1lem  21970  psrass1  22002  psrass23l  22005  psrcom  22006  psrass23  22007  mpfrcl  22127  psropprmul  22255  coe1mul2  22288  lmrcl  23255  1stcrestlem  23476  ptbasfi  23605  isxms2  24474  setsmstopn  24506  tngtopn  24687  rrxmval  25453  ulmss  26455  dchrrcl  27299  gsummpt2co  33034  locfinreflem  33801  sitgclg  34324  cvmsrcl  35249  snmlval  35316  gonan0  35377  bj-fvmptunsn1  37240  eldiophb  42745  elmnc  43125  itgocn  43153  dmmpossx2  48182