MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmmpt Unicode version

Theorem dmmpt 5332
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 5030 . 2  |-  dom  F  =  ran  `' F
2 dfrn4 5298 . 2  |-  ran  `' F  =  ( `' F " _V )
3 dmmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43mptpreima 5330 . 2  |-  ( `' F " _V )  =  { x  e.  A  |  B  e.  _V }
51, 2, 43eqtri 2436 1  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   {crab 2678   _Vcvv 2924    e. cmpt 4234   `'ccnv 4844   dom cdm 4845   ran crn 4846   "cima 4848
This theorem is referenced by:  dmmptss  5333  dmmptg  5334  fvmpti  5772  fvmptss  5780  fvmptss2  5791  tz9.12lem3  7679  cardf2  7794  00lsp  16020  abrexexd  23951  funcnvmptOLD  24043  funcnvmpt  24044  mptctf  24073  issibf  24609  rdgprc0  25372  imageval  25691  dvcosre  27616  itgsinexplem1  27623  stirlinglem14  27711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-mpt 4236  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858
  Copyright terms: Public domain W3C validator