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Theorem dmmpti 5533
Description: Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
fnmpti.1  |-  B  e. 
_V
fnmpti.2  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
dmmpti  |-  dom  F  =  A
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem dmmpti
StepHypRef Expression
1 fnmpti.1 . . 3  |-  B  e. 
_V
2 fnmpti.2 . . 3  |-  F  =  ( x  e.  A  |->  B )
31, 2fnmpti 5532 . 2  |-  F  Fn  A
4 fndm 5503 . 2  |-  ( F  Fn  A  ->  dom  F  =  A )
53, 4ax-mp 8 1  |-  dom  F  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2916    e. cmpt 4226   dom cdm 4837    Fn wfn 5408
This theorem is referenced by:  fvmptex  5774  resfunexg  5916  brtpos2  6444  vdwlem8  13311  dprd2dlem2  15553  dprd2dlem1  15554  dprd2da  15555  ablfac1c  15584  ablfac1eu  15586  ablfaclem2  15599  ablfaclem3  15600  elocv  16850  dfac14  17603  kqtop  17730  symgtgp  18084  eltsms  18115  ressprdsds  18354  minveclem1  19278  isi1f  19519  itg1val  19528  cmvth  19828  mvth  19829  lhop2  19852  dvfsumabs  19860  dvfsumrlim2  19869  taylthlem1  20242  taylthlem2  20243  ulmdvlem1  20269  pige3  20378  relogcn  20482  atandm  20669  atanf  20673  atancn  20729  dmarea  20749  dfarea  20752  efrlim  20761  dchrptlem2  21002  dchrptlem3  21003  dchrisum0  21167  vsfval  22067  ipasslem8  22291  minvecolem1  22329  xppreima2  24013  ofpreima  24034  dmsigagen  24480  measbase  24504  ballotlem7  24746  lgamgulmlem2  24767  eleenn  25739  itg2addnclem2  26156  areacirclem3  26182  totbndbnd  26388  lhe4.4ex1a  27414
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 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-fun 5415  df-fn 5416
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