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Theorem dmmpt 6069
 Description: The domain of the mapping operation in general. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 22-Mar-2015.)
Hypothesis
Ref Expression
dmmpt.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
dmmpt dom 𝐹 = {𝑥𝐴𝐵 ∈ V}

Proof of Theorem dmmpt
StepHypRef Expression
1 dfdm4 5735 . 2 dom 𝐹 = ran 𝐹
2 dfrn4 6031 . 2 ran 𝐹 = (𝐹 “ V)
3 dmmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptpreima 6067 . 2 (𝐹 “ V) = {𝑥𝐴𝐵 ∈ V}
51, 2, 43eqtri 2785 1 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409   ↦ cmpt 5112  ◡ccnv 5523  dom cdm 5524  ran crn 5525   “ cima 5527 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-mpt 5113  df-xp 5530  df-rel 5531  df-cnv 5532  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537 This theorem is referenced by:  dmmptss  6070  dmmptg  6071  dmmptd  6476  fvmpti  6758  fvmptss  6771  fvmptss2  6784  mptexgf  6976  tz9.12lem3  9251  cardf2  9405  pmtrsn  18714  00lsp  19821  rgrx0ndm  27482  abrexexd  30376  funcnvmpt  30528  mptctf  30576  issibf  31819  rdgprc0  33285  imageval  33781  dmmptdf  42222  dmmptssf  42236  dmmptdf2  42237  dvcosre  42920  itgsinexplem1  42962  stirlinglem14  43095  fvmptrabdm  44217
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