Theorem dfrn2 5311

Description: Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)

Assertion

Ref	Expression
dfrn2	|- ran A = { y | E. x x A y }

Distinct variable group:   x, y, A

Proof of Theorem dfrn2

Step	Hyp	Ref	Expression
1		df-rn 5125	. 2 |- ran A = dom `' A
2		df-dm 5124	. 2 |- dom `' A = { y | E. x y `' A x }
3		vex 3203	. . . . 5 |- y e. _V
4		vex 3203	. . . . 5 |- x e. _V
5	3, 4	brcnv 5305	. . . 4 |- ( y `' A x <-> x A y )
6	5	exbii 1774	. . 3 |- ( E. x y `' A x <-> E. x x A y )
7	6	abbii 2739	. 2 |- { y | E. x y `' A x } = { y | E. x x A y }
8	1, 2, 7	3eqtri 2648	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1483   E.wex 1704   {cab 2608   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   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:  dfrn3  5312  dfdm4  5316  dm0rn0  5342  dfrnf  5364  dfima2  5468  funcnv3  5959  opabrn  29424