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Theorem dmmptssf 41861
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 6065 . 2 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
3 dmmptssf.1 . . 3 𝑥𝐴
43ssrab2f 41745 . 2 {𝑥𝐴𝐵 ∈ V} ⊆ 𝐴
52, 4eqsstri 3952 1 dom 𝐹𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2112  wnfc 2939  {crab 3113  Vcvv 3444  wss 3884  cmpt 5113  dom cdm 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-mpt 5114  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536
This theorem is referenced by:  limsupequzmpt2  42353  liminfequzmpt2  42426
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