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Theorem dmmpt 5618
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 5305 . 2 dom 𝐹 = ran 𝐹
2 dfrn4 5583 . 2 ran 𝐹 = (𝐹 “ V)
3 dmmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptpreima 5616 . 2 (𝐹 “ V) = {𝑥𝐴𝐵 ∈ V}
51, 2, 43eqtri 2646 1 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195  cmpt 4720  ccnv 5103  dom cdm 5104  ran crn 5105  cima 5107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-mpt 4721  df-xp 5110  df-rel 5111  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117
This theorem is referenced by:  dmmptss  5619  dmmptg  5620  dmmptd  6011  fvmpti  6268  fvmptss  6279  fvmptss2  6290  mptexgf  6470  tz9.12lem3  8637  cardf2  8754  pmtrsn  17920  00lsp  18962  rgrx0ndm  26470  abrexexd  29319  funcnvmptOLD  29441  funcnvmpt  29442  mptctf  29469  issibf  30369  rdgprc0  31673  imageval  32012  dmmptdf  39233  dmmptssf  39254  dmmptdf2  39255  dvcosre  39889  itgsinexplem1  39932  stirlinglem14  40067
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