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Theorem dmmpti 6010
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 ∈ V
fnmpti.2 𝐹 = (𝑥𝐴𝐵)
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 ∈ V
2 fnmpti.2 . . 3 𝐹 = (𝑥𝐴𝐵)
31, 2fnmpti 6009 . 2 𝐹 Fn 𝐴
4 fndm 5978 . 2 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
53, 4ax-mp 5 1 dom 𝐹 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  wcel 1988  Vcvv 3195  cmpt 4720  dom cdm 5104   Fn wfn 5871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-fun 5878  df-fn 5879
This theorem is referenced by:  fvmptex  6281  resfunexg  6464  brtpos2  7343  vdwlem8  15673  lubdm  16960  glbdm  16973  dprd2dlem2  18420  dprd2dlem1  18421  dprd2da  18422  ablfac1c  18451  ablfac1eu  18453  ablfaclem2  18466  ablfaclem3  18467  elocv  19993  dmtopon  20708  dfac14  21402  kqtop  21529  symgtgp  21886  eltsms  21917  ressprdsds  22157  minveclem1  23176  isi1f  23422  itg1val  23431  cmvth  23735  mvth  23736  lhop2  23759  dvfsumabs  23767  dvfsumrlim2  23776  taylthlem1  24108  taylthlem2  24109  ulmdvlem1  24135  pige3  24250  relogcn  24365  atandm  24584  atanf  24588  atancn  24644  dmarea  24665  dfarea  24668  efrlim  24677  lgamgulmlem2  24737  dchrptlem2  24971  dchrptlem3  24972  dchrisum0  25190  eleenn  25757  incistruhgr  25955  vsfval  27458  ipasslem8  27662  minvecolem1  27700  xppreima2  29423  ofpreima  29439  dmsigagen  30181  measbase  30234  sseqf  30428  ballotlem7  30571  nosupno  31823  nosupdm  31824  nosupbday  31825  nosupres  31827  nosupbnd1lem1  31828  bj-inftyexpidisj  33068  bj-elccinfty  33072  bj-minftyccb  33083  fin2so  33367  poimirlem30  33410  poimir  33413  dvtan  33431  itg2addnclem2  33433  ftc1anclem6  33461  totbndbnd  33559  comptiunov2i  37817  lhe4.4ex1a  38348  dvsinax  39890  fourierdlem62  40148  fourierdlem70  40156  fourierdlem71  40157  fourierdlem80  40166  fouriersw  40211  smflimsuplem1  40789  smflimsuplem4  40792  mndpsuppss  41917  scmsuppss  41918  lincext2  42009  aacllem  42312
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