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Theorem dmmpt 5321
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
dmmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
dmmpt  |-  dom  F  =  { x  e.  A  |  B  e.  _V }

Proof of Theorem dmmpt
StepHypRef Expression
1 dfdm4 5019 . 2  |-  dom  F  =  ran  `' F
2 dfrn4 5286 . 2  |-  ran  `' F  =  ( `' F " _V )
3 dmmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43mptpreima 5319 . 2  |-  ( `' F " _V )  =  { x  e.  A  |  B  e.  _V }
51, 2, 43eqtri 2457 1  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1362    e. wcel 1755   {crab 2709   _Vcvv 2962    e. cmpt 4338   `'ccnv 4826   dom cdm 4827   ran crn 4828   "cima 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-br 4281  df-opab 4339  df-mpt 4340  df-xp 4833  df-rel 4834  df-cnv 4835  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840
This theorem is referenced by:  dmmptss  5322  dmmptg  5323  fvmpti  5761  fvmptss  5770  fvmptss2  5781  tz9.12lem3  7984  cardf2  8101  00lsp  16983  abrexexd  25713  funcnvmptOLD  25809  funcnvmpt  25810  mptctf  25844  issibf  26566  rdgprc0  27453  imageval  27807  dvcosre  29631  itgsinexplem1  29637  stirlinglem14  29725  pmtrsn  30598
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