Theorem elrn2g 5313

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 4403	. . . 4 |- ( y = A -> <. x , y >. = <. x , A >. )
2	1	eleq1d 2686	. . 3 |- ( y = A -> ( <. x , y >. e. B <-> <. x , A >. e. B ) )
3	2	exbidv 1850	. 2 |- ( y = A -> ( E. x <. x , y >. e. B <-> E. x <. x , A >. e. B ) )
4		dfrn3 5312	. 2 |- ran B = { y | E. x <. x , y >. e. B }
5	3, 4	elab2g 3353	1 |- ( A e. V -> ( A e. ran B <-> E. x <. x , A >. e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  elrng  5314  fvrnressn  6428  fo2ndf  7284