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Theorem dmmpt 5081
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
dmmpo.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 4778 . 2  |-  dom  F  =  ran  `' F
2 dfrn4 5046 . 2  |-  ran  `' F  =  ( `' F " _V )
3 dmmpo.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43mptpreima 5079 . 2  |-  ( `' F " _V )  =  { x  e.  A  |  B  e.  _V }
51, 2, 43eqtri 2182 1  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
Colors of variables: wff set class
Syntax hints:    = wceq 1335    e. wcel 2128   {crab 2439   _Vcvv 2712    |-> cmpt 4025   `'ccnv 4585   dom cdm 4586   ran crn 4587   "cima 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-mpt 4027  df-xp 4592  df-rel 4593  df-cnv 4594  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599
This theorem is referenced by:  dmmptss  5082  dmmptg  5083  dmmptd  5300  fvmptssdm  5552  isnumi  7117  dvrecap  13077
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