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Theorem dmmptssf 40355
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 5884 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
3 dmmptssf.1 . . 3 𝑥𝐴
43ssrab2f 40229 . 2 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
52, 4eqsstri 3854 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  wcel 2107  wnfc 2919  {crab 3094  Vcvv 3398  wss 3792  cmpt 4965  dom cdm 5355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-mpt 4966  df-xp 5361  df-rel 5362  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368
This theorem is referenced by:  limsupequzmpt2  40858  liminfequzmpt2  40931
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