Theorem dfrn2 3309

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 3195	. 2 |- ran A = dom `' A
2		df-dm 3194	. 2 |- dom `' A = {y | E.x y`'Ax}
3		visset 1816	. . . . 5 |- y e. V
4		visset 1816	. . . . 5 |- x e. V
5	3, 4	brcnv 3305	. . . 4 |- (y`'Ax <-> xAy)
6	5	exbii 1053	. . 3 |- (E.x y`'Ax <-> E.x xAy)
7	6	abbii 1578	. 2 |- {y | E.x y`'Ax} = {y | E.x xAy}
8	1, 2, 7	3eqtr 1502	1 |- ran A = {y | E.x xAy}

Syntax hints:   = wceq 958  E.wex 982  {cab 1466   class class class wbr 2624  `'ccnv 3175  dom cdm 3176  ran crn 3177

This theorem is referenced by:  dfrn3 3310  dfdm4 3311  dm0rn0 3336  funcnv3 3564  aceq3lem 4742

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-cnv 3192  df-dm 3194  df-rn 3195

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