Theorem elrn2 5019

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 3900	. . . 4 |- ( y = A -> <. x , y >. = <. x , A >. )
3	2	eleq1d 2307	. . 3 |- ( y = A -> ( <. x , y >. e. B <-> <. x , A >. e. B ) )
4	3	exbidv 1878	. 2 |- ( y = A -> ( E. x <. x , y >. e. B <-> E. x <. x , A >. e. B ) )
5		dfrn3 4964	. 2 |- ran B = { y | E. x <. x , y >. e. B }
6	1, 4, 5	elab2 2974	1 |- ( A e. ran B <-> E. x <. x , A >. e. B )

Syntax hints:   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   <.cop 3708   ran crn 4770

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  elrn  5020  dmrnssfld  5040  rniun  5193  rnxpid  5217  ssrnres  5225  relssdmrn  5303