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Theorem dmmpt 4759
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
dmmpt2.1 𝐹 = (x AB)
Assertion
Ref Expression
dmmpt dom 𝐹 = {x AB V}

Proof of Theorem dmmpt
StepHypRef Expression
1 dfdm4 4470 . 2 dom 𝐹 = ran 𝐹
2 dfrn4 4724 . 2 ran 𝐹 = (𝐹 “ V)
3 dmmpt2.1 . . 3 𝐹 = (x AB)
43mptpreima 4757 . 2 (𝐹 “ V) = {x AB V}
51, 2, 43eqtri 2061 1 dom 𝐹 = {x AB V}
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  {crab 2304  Vcvv 2551  cmpt 3809  ccnv 4287  dom cdm 4288  ran crn 4289  cima 4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  dmmptss  4760  dmmptg  4761  fvmptssdm  5198
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