Theorem elrn2g 4801

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

Assertion

Ref	Expression
elrn2g	|- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem elrn2g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 3766	. . . 4 |- ( y = A -> <. x , y >. = <. x , A >. )
2	1	eleq1d 2239	. . 3 |- ( y = A -> ( <. x , y >. e. B <-> <. x , A >. e. B ) )
3	2	exbidv 1818	. 2 |- ( y = A -> ( E. x <. x , y >. e. B <-> E. x <. x , A >. e. B ) )
4		dfrn3 4800	. 2 |- ran B = { y | E. x <. x , y >. e. B }
5	3, 4	elab2g 2877	1 |- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   <.cop 3586   ran crn 4612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by:  elrng  4802  fvelrn  5627  fo2ndf  6206