Theorem dfrn2 3303

Description: Alternate definition of range. Definition 4 of [Suppes] p. 60.

Assertion

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

Distinct variable group:   x,y,A

Proof of Theorem dfrn2

Step	Hyp	Ref	Expression
1		df-rn 3189	. 2 |- ran A = dom `' A
2		df-dm 3188	. 2 |- dom `' A = {y | E.x y`'Ax}
3		visset 1813	. . . . 5 |- y e. V
4		visset 1813	. . . . 5 |- x e. V
5	3, 4	brcnv 3299	. . . 4 |- (y`'Ax <-> xAy)
6	5	exbii 1051	. . 3 |- (E.x y`'Ax <-> E.x xAy)
7	6	abbii 1575	. 2 |- {y | E.x y`'Ax} = {y | E.x xAy}
8	1, 2, 7	3eqtr 1499	1 |- ran A = {y | E.x xAy}

Syntax hints:   = wceq 956  E.wex 980  {cab 1463   class class class wbr 2619  `'ccnv 3169  dom cdm 3170  ran crn 3171

This theorem is referenced by:  dfrn3 3304  dfdm4 3305  dm0rn0 3330  funcnv3 3558  aceq3lem 4732

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-cnv 3186  df-dm 3188  df-rn 3189

Copyright terms: Public domain