Theorem dmmptss 6243

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 6242	. 2 ⊢ dom 𝐹 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
3	2	ssrab3 4044	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913   ↦ cmpt 5196  dom cdm 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-mpt 5197  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675

This theorem is referenced by:  mptrcl  7000  fvmptss  7003  fvmptex  7005  fvmptnf  7013  elfvmptrab1w  7018  elfvmptrab1  7019  mptexg  7220  mptexw  7950  dmmpossx  8063  tposssxp  8226  mptfi  9308  cnvimamptfin  9310  cantnfres  9646  mptct  10522  arwrcl  18101  submgmrcl  18753  cntzrcl  19397  gsumconst  20004  psrass1lem  22052  psrass1  22082  psrass23l  22085  psrcom  22086  psrass23  22087  mpfrcl  22205  psropprmul  22366  coe1mul2  22399  lmrcl  23357  1stcrestlem  23578  ptbasfi  23707  isxms2  24574  setsmstopn  24604  tngtopn  24776  rrxmval  25533  ulmss  26526  dchrrcl  27370  gsummpt2co  33309  locfinreflem  34175  sitgclg  34677  cvmsrcl  35655  snmlval  35722  gonan0  35783  bj-fvmptunsn1  37789  eldiophb  43380  elmnc  43755  itgocn  43783  tannpoly  47516  dmmpossx2  49002  dmtposss  49539