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Theorem rnex 4855
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 4853 . 2 (𝐴 ∈ V → ran 𝐴 ∈ V)
31, 2ax-mp 5 1 ran 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 2128  Vcvv 2712  ran crn 4589
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-cnv 4596  df-dm 4598  df-rn 4599
This theorem is referenced by:  ffoss  5448  abrexex  6067  fo2nd  6108  tfrexlem  6283  ixpsnf1o  6683  bren  6694  xpassen  6777  mapen  6793  ssenen  6798  seqex  10355  hashfacen  10718  shftfval  10732  restfn  12425  tgioo  13016
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