Theorem rnex 5000

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 4997	. 2 |- ( A e. _V -> ran A e. _V )
3	1, 2	ax-mp 5	1 |- ran A e. _V

Syntax hints:   e. wcel 2202   _Vcvv 2802   ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  ffoss  5616  abrexex  6279  fo2nd  6321  tfrexlem  6500  ixpsnf1o  6905  bren  6917  xpassen  7014  mapen  7032  ssenen  7037  seqex  10711  hashfacen  11100  shftfval  11382  restfn  13327  prdsvallem  13356  prdsval  13357  mopnset  14568  metuex  14571  tgioo  15280