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Theorem dmmpt 5042
 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
dmmpo.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
dmmpt dom 𝐹 = {𝑥𝐴𝐵 ∈ V}

Proof of Theorem dmmpt
StepHypRef Expression
1 dfdm4 4739 . 2 dom 𝐹 = ran 𝐹
2 dfrn4 5007 . 2 ran 𝐹 = (𝐹 “ V)
3 dmmpo.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43mptpreima 5040 . 2 (𝐹 “ V) = {𝑥𝐴𝐵 ∈ V}
51, 2, 43eqtri 2165 1 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ∈ wcel 1481  {crab 2421  Vcvv 2689   ↦ cmpt 3997  ◡ccnv 4546  dom cdm 4547  ran crn 4548   “ cima 4550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-mpt 3999  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560 This theorem is referenced by:  dmmptss  5043  dmmptg  5044  dmmptd  5261  fvmptssdm  5513  isnumi  7056  dvrecap  12886
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