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Theorem dmmpt 5184
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 4888 . 2  |-  dom  F  =  ran  `' F
2 dfrn4 5150 . 2  |-  ran  `' F  =  ( `' F " _V )
3 dmmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43mptpreima 5182 . 2  |-  ( `' F " _V )  =  { x  e.  A  |  B  e.  _V }
51, 2, 43eqtri 2320 1  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708
This theorem is referenced by:  dmmptss  5185  dmmptg  5186  fvmpti  5617  fvmptss  5625  fvmptss2  5635  tz9.12lem3  7477  cardf2  7592  00lsp  15754  abrexexd  23207  funcnvmptOLD  23249  funcnvmpt  23250  mptctf  23363  rdgprc0  24221  imageval  24540  dvcosre  27844  itgsinexplem1  27851  stirlinglem14  27939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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