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Theorem dmmpt 6239
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 5895 . 2 dom 𝐹 = ran 𝐹
2 dfrn4 6201 . 2 ran 𝐹 = (𝐹 “ V)
3 dmmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptpreima 6237 . 2 (𝐹 “ V) = {𝑥𝐴𝐵 ∈ V}
51, 2, 43eqtri 2763 1 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2105  {crab 3431  Vcvv 3473  cmpt 5231  ccnv 5675  dom cdm 5676  ran crn 5677  cima 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  dmmptss  6240  dmmptg  6241  dmmptd  6695  fvmpti  6997  fvmptss  7010  fvmptss2  7023  mptexgf  7226  tz9.12lem3  9790  cardf2  9944  pmtrsn  19435  00lsp  20824  rgrx0ndm  29284  abrexexd  32180  funcnvmpt  32326  mptctf  32376  issibf  33797  rdgprc0  35236  imageval  35373  dmmptdff  44383  dmmptssf  44395  dmmptdf2  44396  dvcosre  45089  itgsinexplem1  45131  stirlinglem14  45264  fvmptrabdm  46462
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