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Theorem dmmpti 5710
Description: Domain of the mapping operation. (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 5709 . 2  |-  F  Fn  A
4 fndm 5680 . 2  |-  ( F  Fn  A  ->  dom  F  =  A )
53, 4ax-mp 5 1  |-  dom  F  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505   dom cdm 4999    Fn wfn 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-fun 5590  df-fn 5591
This theorem is referenced by:  fvmptex  5960  resfunexg  6126  brtpos2  6961  vdwlem8  14365  lubdm  15466  glbdm  15479  dprd2dlem2  16891  dprd2dlem1  16892  dprd2da  16893  ablfac1c  16924  ablfac1eu  16926  ablfaclem2  16939  ablfaclem3  16940  elocv  18494  dfac14  19882  kqtop  20009  symgtgp  20363  eltsms  20394  ressprdsds  20637  minveclem1  21602  isi1f  21844  itg1val  21853  cmvth  22155  mvth  22156  lhop2  22179  dvfsumabs  22187  dvfsumrlim2  22196  taylthlem1  22530  taylthlem2  22531  ulmdvlem1  22557  pige3  22671  relogcn  22775  atandm  22963  atanf  22967  atancn  23023  dmarea  23043  dfarea  23046  efrlim  23055  dchrptlem2  23296  dchrptlem3  23297  dchrisum0  23461  eleenn  23903  vsfval  25232  ipasslem8  25456  minvecolem1  25494  xppreima2  27188  ofpreima  27207  dmsigagen  27812  measbase  27836  ballotlem7  28142  lgamgulmlem2  28240  fin2so  29645  dvtan  29670  itg2addnclem2  29672  ftc1anclem6  29700  totbndbnd  29916  lhe4.4ex1a  30862  dvsinax  31269  fourierdlem62  31497  fourierdlem71  31506  fourierdlem80  31515  fouriersw  31560  mndpsuppss  32063  scmsuppss  32064  lincext2  32155  bj-inftyexpidisj  33703  bj-elccinfty  33707  bj-minftyccb  33718
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