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Theorem dmmpt 6092
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 5763 . 2 dom 𝐹 = ran 𝐹
2 dfrn4 6057 . 2 ran 𝐹 = (𝐹 “ V)
3 dmmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptpreima 6090 . 2 (𝐹 “ V) = {𝑥𝐴𝐵 ∈ V}
51, 2, 43eqtri 2853 1 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2107  {crab 3147  Vcvv 3500  cmpt 5143  ccnv 5553  dom cdm 5554  ran crn 5555  cima 5557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-mpt 5144  df-xp 5560  df-rel 5561  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567
This theorem is referenced by:  dmmptss  6093  dmmptg  6094  dmmptd  6490  fvmpti  6764  fvmptss  6776  fvmptss2  6789  mptexgf  6980  tz9.12lem3  9207  cardf2  9361  pmtrsn  18567  00lsp  19673  rgrx0ndm  27289  abrexexd  30183  funcnvmpt  30327  mptctf  30366  issibf  31477  rdgprc0  32922  imageval  33275  dmmptdf  41353  dmmptssf  41367  dmmptdf2  41368  dvcosre  42061  itgsinexplem1  42104  stirlinglem14  42238  fvmptrabdm  43358
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