Theorem dfima2 5126

Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dfima2	|- ( A " B ) = { y | E. x e. B x A y }

Distinct variable groups:   x, y, A   x, B, y

Proof of Theorem dfima2

Step	Hyp	Ref	Expression
1		df-ima 4785	. 2 |- ( A " B ) = ran ( A |` B )
2		dfrn2 4966	. 2 |- ran ( A |` B ) = { y | E. x x ( A |` B ) y }
3		vex 2824	. . . . . . 7 |- y e. _V
4	3	brres 5067	. . . . . 6 |- ( x ( A |` B ) y <-> ( x A y /\ x e. B ) )
5		ancom 266	. . . . . 6 |- ( ( x A y /\ x e. B ) <-> ( x e. B /\ x A y ) )
6	4, 5	bitri 184	. . . . 5 |- ( x ( A |` B ) y <-> ( x e. B /\ x A y ) )
7	6	exbii 1658	. . . 4 |- ( E. x x ( A |` B ) y <-> E. x ( x e. B /\ x A y ) )
8		df-rex 2534	. . . 4 |- ( E. x e. B x A y <-> E. x ( x e. B /\ x A y ) )
9	7, 8	bitr4i 187	. . 3 |- ( E. x x ( A |` B ) y <-> E. x e. B x A y )
10	9	abbii 2354	. 2 |- { y | E. x x ( A |` B ) y } = { y | E. x e. B x A y }
11	1, 2, 10	3eqtri 2263	1 |- ( A " B ) = { y | E. x e. B x A y }

Syntax hints:   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   E.wrex 2529   class class class wbr 4128   ran crn 4773   |` cres 4774   "cima 4775

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785

This theorem is referenced by:  dfima3  5127  elimag  5128  imasng  5150  imadiflem  5458  imadif  5459  imainlem  5460  imain  5461  funimaexglem  5462  dfimafn  5748  dfimafnf  5948  isoini  6017