Theorem elrng 4951

Description: Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem elrng

Step	Hyp	Ref	Expression
1		elrn2g 4950	. 2 |- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )
2		df-br 4115	. . 3 |- ( x B A <-> <. x , A >. e. B )
3	2	exbii 1654	. 2 |- ( E. x x B A <-> E. x <. x , A >. e. B )
4	1, 3	bitr4di 198	1 |- ( A e. V -> ( A e. ran B <-> E. x x B A ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1541   e. wcel 2205   <.cop 3697   class class class wbr 4114   ran crn 4755

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by:  ssrelrn  4952  relelrnb  5000  wlkvtxiedg  16466  wlkvtxiedgg  16467