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Theorem dfima3 5198
Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dfima3  |-  ( A
" B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  A
) }
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem dfima3
StepHypRef Expression
1 dfima2 5197 . 2  |-  ( A
" B )  =  { y  |  E. x  e.  B  x A y }
2 df-br 4205 . . . . 5  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
32rexbii 2722 . . . 4  |-  ( E. x  e.  B  x A y  <->  E. x  e.  B  <. x ,  y >.  e.  A
)
4 df-rex 2703 . . . 4  |-  ( E. x  e.  B  <. x ,  y >.  e.  A  <->  E. x ( x  e.  B  /\  <. x ,  y >.  e.  A
) )
53, 4bitri 241 . . 3  |-  ( E. x  e.  B  x A y  <->  E. x
( x  e.  B  /\  <. x ,  y
>.  e.  A ) )
65abbii 2547 . 2  |-  { y  |  E. x  e.  B  x A y }  =  { y  |  E. x ( x  e.  B  /\  <.
x ,  y >.  e.  A ) }
71, 6eqtri 2455 1  |-  ( A
" B )  =  { y  |  E. x ( x  e.  B  /\  <. x ,  y >.  e.  A
) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   <.cop 3809   class class class wbr 4204   "cima 4873
This theorem is referenced by:  imadmrn  5207  imassrn  5208  imai  5210  funimaexg  5522  rdglim2  6682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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