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Theorem dmmpt 5774
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 5454 . 2 dom 𝐹 = ran 𝐹
2 dfrn4 5736 . 2 ran 𝐹 = (𝐹 “ V)
3 dmmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptpreima 5772 . 2 (𝐹 “ V) = {𝑥𝐴𝐵 ∈ V}
51, 2, 43eqtri 2796 1 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1630  wcel 2144  {crab 3064  Vcvv 3349  cmpt 4861  ccnv 5248  dom cdm 5249  ran crn 5250  cima 5252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-mpt 4862  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262
This theorem is referenced by:  dmmptss  5775  dmmptg  5776  dmmptd  6164  fvmpti  6423  fvmptss  6434  fvmptss2  6446  mptexgf  6628  tz9.12lem3  8815  cardf2  8968  pmtrsn  18145  00lsp  19193  rgrx0ndm  26723  abrexexd  29679  funcnvmptOLD  29801  funcnvmpt  29802  mptctf  29829  issibf  30729  rdgprc0  32029  imageval  32368  dmmptdf  39929  dmmptssf  39950  dmmptdf2  39951  dvcosre  40638  itgsinexplem1  40681  stirlinglem14  40815  fvmptrabdm  41825
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