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Theorem dmmpt 6240
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
dmmpt.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
dmmpt dom 𝐹 = {𝑥𝐴𝐵 ∈ V}

Proof of Theorem dmmpt
StepHypRef Expression
1 dfdm4 5896 . 2 dom 𝐹 = ran 𝐹
2 dfrn4 6202 . 2 ran 𝐹 = (𝐹 “ V)
3 dmmpt.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptpreima 6238 . 2 (𝐹 “ V) = {𝑥𝐴𝐵 ∈ V}
51, 2, 43eqtri 2762 1 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2104  {crab 3430  Vcvv 3472  cmpt 5232  ccnv 5676  dom cdm 5677  ran crn 5678  cima 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690
This theorem is referenced by:  dmmptss  6241  dmmptg  6242  dmmptd  6696  fvmpti  6998  fvmptss  7011  fvmptss2  7024  mptexgf  7227  tz9.12lem3  9788  cardf2  9942  pmtrsn  19430  00lsp  20738  rgrx0ndm  29115  abrexexd  32011  funcnvmpt  32157  mptctf  32207  issibf  33628  rdgprc0  35067  imageval  35204  dmmptdff  44222  dmmptssf  44234  dmmptdf2  44235  dvcosre  44928  itgsinexplem1  44970  stirlinglem14  45103  fvmptrabdm  46301
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