Theorem dfrn3 5312

Description: Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Assertion

Ref	Expression
dfrn3	|- ran A = { y | E. x <. x , y >. e. A }

Distinct variable group:   x, y, A

Proof of Theorem dfrn3

Step	Hyp	Ref	Expression
1		dfrn2 5311	. 2 |- ran A = { y | E. x x A y }
2		df-br 4654	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1774	. . 3 |- ( E. x x A y <-> E. x <. x , y >. e. A )
4	3	abbii 2739	. 2 |- { y | E. x x A y } = { y | E. x <. x , y >. e. A }
5	1, 4	eqtri 2644	1 |- ran A = { y | E. x <. x , y >. e. A }

Syntax hints:   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   <.cop 4183   class class class wbr 4653   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:  elrn2g  5313  elrn2  5365  imadmrn  5476  imassrn  5477  csbrngOLD  39056  csbrngVD  39132