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Theorem dmmpt 6262
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 5909 . 2 dom 𝐹 = ran 𝐹
2 dfrn4 6224 . 2 ran 𝐹 = (𝐹 “ V)
3 dmmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptpreima 6260 . 2 (𝐹 “ V) = {𝑥𝐴𝐵 ∈ V}
51, 2, 43eqtri 2767 1 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cmpt 5231  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  dmmptss  6263  dmmptg  6264  dmmptd  6714  fvmpti  7015  fvmptss  7028  fvmptss2  7042  mptexgf  7242  tz9.12lem3  9827  cardf2  9981  pmtrsn  19552  00lsp  20997  rgrx0ndm  29626  abrexexd  32537  funcnvmpt  32684  mptctf  32735  issibf  34315  rdgprc0  35775  imageval  35912  dmmptdff  45166  dmmptssf  45175  dmmptdf2  45176  dvcosre  45868  itgsinexplem1  45910  stirlinglem14  46043  fvmptrabdm  47243
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