Theorem dfrn2 4918

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 4736	. 2 |- ran A = dom `' A
2		df-dm 4735	. 2 |- dom `' A = { y | E. x y `' A x }
3		vex 2805	. . . . 5 |- y e. _V
4		vex 2805	. . . . 5 |- x e. _V
5	3, 4	brcnv 4913	. . . 4 |- ( y `' A x <-> x A y )
6	5	exbii 1653	. . 3 |- ( E. x y `' A x <-> E. x x A y )
7	6	abbii 2347	. 2 |- { y | E. x y `' A x } = { y | E. x x A y }
8	1, 2, 7	3eqtri 2256	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1397   E.wex 1540   {cab 2217   class class class wbr 4088   `'ccnv 4724   dom cdm 4725   ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  dfrn3  4919  dfdm4  4923  dm0rn0  4948  dmmrnm  4951  dfrnf  4973  dfima2  5078  funcnv3  5392