Theorem elrng 4853

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 4852	. 2 |- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )
2		df-br 4030	. . 3 |- ( x B A <-> <. x , A >. e. B )
3	2	exbii 1616	. 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 1503   e. wcel 2164   <.cop 3621   class class class wbr 4029   ran crn 4660

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by:  relelrnb  4900