Theorem elrn 5000

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 4999	. 2 |- ( A e. ran B <-> E. x <. x , A >. e. B )
3		df-br 4110	. . 3 |- ( x B A <-> <. x , A >. e. B )
4	3	exbii 1654	. 2 |- ( E. x x B A <-> E. x <. x , A >. e. B )
5	2, 4	bitr4i 187	1 |- ( A e. ran B <-> E. x x B A )

Syntax hints:   <-> wb 105   E.wex 1541   e. wcel 2203   _Vcvv 2813   <.cop 3692   class class class wbr 4109   ran crn 4750

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by:  dmcosseq  5029  rnco  5269  dffo4  5825  rntpos  6488  fclim  11979  dvfgg  15553