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Theorem dmmpt 5444
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 5143 . 2  |-  dom  F  =  ran  `' F
2 dfrn4 5409 . 2  |-  ran  `' F  =  ( `' F " _V )
3 dmmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43mptpreima 5442 . 2  |-  ( `' F " _V )  =  { x  e.  A  |  B  e.  _V }
51, 2, 43eqtri 2487 1  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   {crab 2803   _Vcvv 3078    |-> cmpt 4461   `'ccnv 4950   dom cdm 4951   ran crn 4952   "cima 4954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-mpt 4463  df-xp 4957  df-rel 4958  df-cnv 4959  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964
This theorem is referenced by:  dmmptss  5445  dmmptg  5446  fvmpti  5885  fvmptss  5894  fvmptss2  5905  tz9.12lem3  8111  cardf2  8228  pmtrsn  16148  00lsp  17195  abrexexd  26069  funcnvmptOLD  26164  funcnvmpt  26165  mptctf  26199  issibf  26886  rdgprc0  27774  imageval  28128  dvcosre  29959  itgsinexplem1  29965  stirlinglem14  30053
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