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Theorem dmmptss 4845
Description: The domain of a mapping is a subset of its base class. (Contributed by Scott Fenton, 17-Jun-2013.)
Hypothesis
Ref Expression
dmmpt2.1 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 dmmpt2.1 . . 3 𝐹 = (𝑥𝐴𝐵)
21dmmpt 4844 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
3 ssrab2 3053 . 2 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
42, 3eqsstri 3003 1 dom 𝐹𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  {crab 2327  Vcvv 2574  wss 2945  cmpt 3846  dom cdm 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-mpt 3848  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386
This theorem is referenced by:  fvmptssdm  5283  mptexg  5414  dmmpt2ssx  5853  tposssxp  5895
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