ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rnex GIF version

Theorem rnex 4624
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 4622 . 2 (𝐴 ∈ V → ran 𝐴 ∈ V)
31, 2ax-mp 7 1 ran 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 1407  Vcvv 2572  ran crn 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-cnv 4378  df-dm 4380  df-rn 4381
This theorem is referenced by:  ffoss  5183  abrexex  5769  fo2nd  5810  tfrexlem  5976  bren  6256  xpassen  6332  iseqex  9342  shftfval  9614
  Copyright terms: Public domain W3C validator