Theorem dfima2 5076

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 4736	. 2 |- ( A " B ) = ran ( A |` B )
2		dfrn2 4916	. 2 |- ran ( A |` B ) = { y | E. x x ( A |` B ) y }
3		vex 2803	. . . . . . 7 |- y e. _V
4	3	brres 5017	. . . . . 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 1651	. . . 4 |- ( E. x x ( A |` B ) y <-> E. x ( x e. B /\ x A y ) )
8		df-rex 2514	. . . 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 2345	. 2 |- { y | E. x x ( A |` B ) y } = { y | E. x e. B x A y }
11	1, 2, 10	3eqtri 2254	1 |- ( A " B ) = { y | E. x e. B x A y }

Syntax hints:   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   E.wrex 2509   class class class wbr 4086   ran crn 4724   |` cres 4725   "cima 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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736

This theorem is referenced by:  dfima3  5077  elimag  5078  imasng  5099  imadiflem  5406  imadif  5407  imainlem  5408  imain  5409  funimaexglem  5410  dfimafn  5690  isoini  5954