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Theorem dmmptssf 45237
Description: The domain of a mapping is a subset of its base class. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
dmmptssf.1 𝑥𝐴
dmmptssf.2 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
dmmptssf dom 𝐹𝐴

Proof of Theorem dmmptssf
StepHypRef Expression
1 dmmptssf.2 . . 3 𝐹 = (𝑥𝐴𝐵)
21dmmpt 6260 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
3 dmmptssf.1 . . 3 𝑥𝐴
43ssrab2f 45122 . 2 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
52, 4eqsstri 4030 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  wnfc 2890  {crab 3436  Vcvv 3480  wss 3951  cmpt 5225  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by:  rn1st  45280  limsupequzmpt2  45733  liminfequzmpt2  45806
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