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Theorem dmmptss 5003
 Description: The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
Hypothesis
Ref Expression
dmmpo.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 dmmpo.1 . . 3 𝐹 = (𝑥𝐴𝐵)
21dmmpt 5002 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
3 ssrab2 3150 . 2 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
42, 3eqsstri 3097 1 dom 𝐹𝐴
 Colors of variables: wff set class Syntax hints:   = wceq 1314   ∈ wcel 1463  {crab 2395  Vcvv 2658   ⊆ wss 3039   ↦ cmpt 3957  dom cdm 4507 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-mpt 3959  df-xp 4513  df-rel 4514  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520 This theorem is referenced by:  mptrcl  5469  fvmptssdm  5471  elfvmptrab1  5481  mptexg  5611  dmmpossx  6063  tposssxp  6112  lmrcl  12266  cnprcl2k  12281  isxms2  12527
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