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Theorem rnex 4878
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 4876 . 2 (𝐴 ∈ V → ran 𝐴 ∈ V)
31, 2ax-mp 5 1 ran 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 2141  Vcvv 2730  ran crn 4612
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622
This theorem is referenced by:  ffoss  5474  abrexex  6096  fo2nd  6137  tfrexlem  6313  ixpsnf1o  6714  bren  6725  xpassen  6808  mapen  6824  ssenen  6829  seqex  10403  hashfacen  10771  shftfval  10785  restfn  12583  tgioo  13340
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