Theorem rnex 4992

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 e. _V

Assertion

Ref	Expression
rnex	|- ran A e. _V

Proof of Theorem rnex

Step	Hyp	Ref	Expression
1		dmex.1	. 2 |- A e. _V
2		rnexg 4989	. 2 |- ( A e. _V -> ran A e. _V )
3	1, 2	ax-mp 5	1 |- ran A e. _V

Syntax hints:   e. wcel 2200   _Vcvv 2799   ran crn 4720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  ffoss  5604  abrexex  6262  fo2nd  6304  tfrexlem  6480  ixpsnf1o  6883  bren  6895  xpassen  6989  mapen  7007  ssenen  7012  seqex  10671  hashfacen  11058  shftfval  11332  restfn  13276  prdsvallem  13305  prdsval  13306  mopnset  14516  metuex  14519  tgioo  15228