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Theorem rnex 4713
Description: The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.)
Hypothesis
Ref Expression
dmex.1 𝐴 ∈ V
Assertion
Ref Expression
rnex ran 𝐴 ∈ V

Proof of Theorem rnex
StepHypRef Expression
1 dmex.1 . 2 𝐴 ∈ V
2 rnexg 4711 . 2 (𝐴 ∈ V → ran 𝐴 ∈ V)
31, 2ax-mp 7 1 ran 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 1439  Vcvv 2620  ran crn 4453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-cnv 4460  df-dm 4462  df-rn 4463
This theorem is referenced by:  ffoss  5298  abrexex  5902  fo2nd  5943  tfrexlem  6113  ixpsnf1o  6507  bren  6518  xpassen  6600  mapen  6616  ssenen  6621  iseqex  9917  hashfacen  10302  shftfval  10316  restfn  11717
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