Theorem elrn2g 4729

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 3706	. . . 4 |- ( y = A -> <. x , y >. = <. x , A >. )
2	1	eleq1d 2208	. . 3 |- ( y = A -> ( <. x , y >. e. B <-> <. x , A >. e. B ) )
3	2	exbidv 1797	. 2 |- ( y = A -> ( E. x <. x , y >. e. B <-> E. x <. x , A >. e. B ) )
4		dfrn3 4728	. 2 |- ran B = { y | E. x <. x , y >. e. B }
5	3, 4	elab2g 2831	1 |- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   <.cop 3530   ran crn 4540

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550

This theorem is referenced by:  elrng  4730  fvelrn  5551  fo2ndf  6124