Theorem elrn2g 4794

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 3759	. . . 4 |- ( y = A -> <. x , y >. = <. x , A >. )
2	1	eleq1d 2235	. . 3 |- ( y = A -> ( <. x , y >. e. B <-> <. x , A >. e. B ) )
3	2	exbidv 1813	. 2 |- ( y = A -> ( E. x <. x , y >. e. B <-> E. x <. x , A >. e. B ) )
4		dfrn3 4793	. 2 |- ran B = { y | E. x <. x , y >. e. B }
5	3, 4	elab2g 2873	1 |- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   <.cop 3579   ran crn 4605

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  elrng  4795  fvelrn  5616  fo2ndf  6195