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Theorem dmmpt 5594
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 5281 . 2 dom 𝐹 = ran 𝐹
2 dfrn4 5559 . 2 ran 𝐹 = (𝐹 “ V)
3 dmmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptpreima 5592 . 2 (𝐹 “ V) = {𝑥𝐴𝐵 ∈ V}
51, 2, 43eqtri 2647 1 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3189  cmpt 4678  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-mpt 4680  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092
This theorem is referenced by:  dmmptss  5595  dmmptg  5596  dmmptd  5986  fvmpti  6243  fvmptss  6254  fvmptss2  6265  mptexgf  6445  tz9.12lem3  8604  cardf2  8721  pmtrsn  17871  00lsp  18913  rgrx0ndm  26376  abrexexd  29217  funcnvmptOLD  29333  funcnvmpt  29334  mptctf  29361  issibf  30200  rdgprc0  31435  imageval  31714  dmmptdf  38922  dmmptssf  38944  dmmptdf2  38945  dvcosre  39457  itgsinexplem1  39502  stirlinglem14  39637
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