Theorem elrn2 5001

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 3886	. . . 4 |- ( y = A -> <. x , y >. = <. x , A >. )
3	2	eleq1d 2303	. . 3 |- ( y = A -> ( <. x , y >. e. B <-> <. x , A >. e. B ) )
4	3	exbidv 1874	. 2 |- ( y = A -> ( E. x <. x , y >. e. B <-> E. x <. x , A >. e. B ) )
5		dfrn3 4946	. 2 |- ran B = { y | E. x <. x , y >. e. B }
6	1, 4, 5	elab2 2967	1 |- ( A e. ran B <-> E. x <. x , A >. e. B )

Syntax hints:   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815   <.cop 3694   ran crn 4752

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-cnv 4759  df-dm 4761  df-rn 4762

This theorem is referenced by:  elrn  5002  dmrnssfld  5022  rniun  5175  rnxpid  5199  ssrnres  5207  relssdmrn  5285