Theorem elrn 5366

Description: Membership in a range. (Contributed by NM, 2-Apr-2004.)

Hypothesis

Ref	Expression
elrn.1	|- A e. _V

Assertion

Ref	Expression
elrn	|- ( A e. ran B <-> E. x x B A )

Distinct variable groups:   x, A   x, B

Proof of Theorem elrn

Step	Hyp	Ref	Expression
1		elrn.1	. . 3 |- A e. _V
2	1	elrn2 5365	. 2 |- ( A e. ran B <-> E. x <. x , A >. e. B )
3		df-br 4654	. . 3 |- ( x B A <-> <. x , A >. e. B )
4	3	exbii 1774	. 2 |- ( E. x x B A <-> E. x <. x , A >. e. B )
5	2, 4	bitr4i 267	1 |- ( A e. ran B <-> E. x x B A )

Syntax hints:   <-> wb 196   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  dmcosseq  5387  inisegn0  5497  rnco  5641  dffo4  6375  fvclss  6500  rntpos  7365  fpwwe2lem11  9462  fpwwe2lem12  9463  fclim  14284  perfdvf  23667  dftr6  31640  dffr5  31643  brsset  31996  dfon3  31999  brtxpsd  32001  dffix2  32012  elsingles  32025  dfrdg4  32058  undmrnresiss  37910