Theorem elrn2 4972

Description: Membership in a range. (Contributed by NM, 10-Jul-1994.)

Hypothesis

Ref	Expression
elrn.1	|- A e. _V

Assertion

Ref	Expression
elrn2	|- ( A e. ran B <-> E. x <. x , A >. e. B )

Distinct variable groups:   x, A   x, B

Proof of Theorem elrn2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrn.1	. 2 |- A e. _V
2		opeq2 3861	. . . 4 |- ( y = A -> <. x , y >. = <. x , A >. )
3	2	eleq1d 2298	. . 3 |- ( y = A -> ( <. x , y >. e. B <-> <. x , A >. e. B ) )
4	3	exbidv 1871	. 2 |- ( y = A -> ( E. x <. x , y >. e. B <-> E. x <. x , A >. e. B ) )
5		dfrn3 4917	. 2 |- ran B = { y | E. x <. x , y >. e. B }
6	1, 4, 5	elab2 2952	1 |- ( A e. ran B <-> E. x <. x , A >. e. B )

Syntax hints:   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2800   <.cop 3670   ran crn 4724

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-cnv 4731  df-dm 4733  df-rn 4734

This theorem is referenced by:  elrn  4973  dmrnssfld  4993  rniun  5145  rnxpid  5169  ssrnres  5177  relssdmrn  5255