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Theorem dmmpt 5368
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 5066 . 2  |-  dom  F  =  ran  `' F
2 dfrn4 5334 . 2  |-  ran  `' F  =  ( `' F " _V )
3 dmmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43mptpreima 5366 . 2  |-  ( `' F " _V )  =  { x  e.  A  |  B  e.  _V }
51, 2, 43eqtri 2462 1  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    e. cmpt 4269   `'ccnv 4880   dom cdm 4881   ran crn 4882   "cima 4884
This theorem is referenced by:  dmmptss  5369  dmmptg  5370  fvmpti  5808  fvmptss  5816  fvmptss2  5827  tz9.12lem3  7718  cardf2  7835  00lsp  16062  abrexexd  23995  funcnvmptOLD  24087  funcnvmpt  24088  mptctf  24117  issibf  24653  rdgprc0  25426  imageval  25780  dvcosre  27731  itgsinexplem1  27738  stirlinglem14  27826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-mpt 4271  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894
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