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Theorem dmmpt 5508
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 5205 . 2  |-  dom  F  =  ran  `' F
2 dfrn4 5473 . 2  |-  ran  `' F  =  ( `' F " _V )
3 dmmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43mptpreima 5506 . 2  |-  ( `' F " _V )  =  { x  e.  A  |  B  e.  _V }
51, 2, 43eqtri 2490 1  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-mpt 4517  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021
This theorem is referenced by:  dmmptss  5509  dmmptg  5510  dmmptd  5717  fvmpti  5955  fvmptss  5965  fvmptss2  5976  tz9.12lem3  8224  cardf2  8341  pmtrsn  16671  00lsp  17754  abrexexd  27535  mptexgf  27614  funcnvmptOLD  27663  funcnvmpt  27664  mptctf  27701  issibf  28472  rdgprc0  29443  imageval  29785  dvcosre  31909  itgsinexplem1  31955  stirlinglem14  32072
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