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Theorem dmmpti 5373
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 5372 . 2  |-  F  Fn  A
4 fndm 5343 . 2  |-  ( F  Fn  A  ->  dom  F  =  A )
53, 4ax-mp 8 1  |-  dom  F  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   dom cdm 4689    Fn wfn 5250
This theorem is referenced by:  fvmptex  5610  resfunexg  5737  brtpos2  6240  vdwlem8  13035  dprd2dlem2  15275  dprd2dlem1  15276  dprd2da  15277  ablfac1c  15306  ablfac1eu  15308  ablfaclem2  15321  ablfaclem3  15322  elocv  16568  dfac14  17312  kqtop  17436  symgtgp  17784  eltsms  17815  ressprdsds  17935  minveclem1  18788  isi1f  19029  itg1val  19038  cmvth  19338  mvth  19339  lhop2  19362  dvfsumabs  19370  dvfsumrlim2  19379  taylthlem1  19752  taylthlem2  19753  ulmdvlem1  19777  pige3  19885  relogcn  19985  atandm  20172  atanf  20176  atancn  20232  dmarea  20252  dfarea  20255  efrlim  20264  dchrptlem2  20504  dchrptlem3  20505  dchrisum0  20669  vsfval  21191  ipasslem8  21415  minvecolem1  21453  ballotlem7  23094  dmsigagen  23505  eleenn  24524  areacirclem3  24926  cmpdom2  25144  cndpv  26049  totbndbnd  26513  lhe4.4ex1a  27546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
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