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Theorem rnco 5216
Description: The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
Assertion
Ref Expression
rnco  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )

Proof of Theorem rnco
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2825 . . . . . 6  |-  x  e. 
_V
2 vex 2825 . . . . . 6  |-  y  e. 
_V
31, 2brco 4889 . . . . 5  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
43exbii 1573 . . . 4  |-  ( E. x  x ( A  o.  B ) y  <->  E. x E. z ( x B z  /\  z A y ) )
5 excom 1817 . . . 4  |-  ( E. x E. z ( x B z  /\  z A y )  <->  E. z E. x ( x B z  /\  z A y ) )
6 ancom 437 . . . . . . 7  |-  ( ( E. x  x B z  /\  z A y )  <->  ( z A y  /\  E. x  x B z ) )
7 19.41v 1873 . . . . . . 7  |-  ( E. x ( x B z  /\  z A y )  <->  ( E. x  x B z  /\  z A y ) )
8 vex 2825 . . . . . . . . 9  |-  z  e. 
_V
98elrn 4956 . . . . . . . 8  |-  ( z  e.  ran  B  <->  E. x  x B z )
109anbi2i 675 . . . . . . 7  |-  ( ( z A y  /\  z  e.  ran  B )  <-> 
( z A y  /\  E. x  x B z ) )
116, 7, 103bitr4i 268 . . . . . 6  |-  ( E. x ( x B z  /\  z A y )  <->  ( z A y  /\  z  e.  ran  B ) )
122brres 4998 . . . . . 6  |-  ( z ( A  |`  ran  B
) y  <->  ( z A y  /\  z  e.  ran  B ) )
1311, 12bitr4i 243 . . . . 5  |-  ( E. x ( x B z  /\  z A y )  <->  z ( A  |`  ran  B ) y )
1413exbii 1573 . . . 4  |-  ( E. z E. x ( x B z  /\  z A y )  <->  E. z 
z ( A  |`  ran  B ) y )
154, 5, 143bitri 262 . . 3  |-  ( E. x  x ( A  o.  B ) y  <->  E. z  z ( A  |`  ran  B ) y )
162elrn 4956 . . 3  |-  ( y  e.  ran  ( A  o.  B )  <->  E. x  x ( A  o.  B ) y )
172elrn 4956 . . 3  |-  ( y  e.  ran  ( A  |`  ran  B )  <->  E. z 
z ( A  |`  ran  B ) y )
1815, 16, 173bitr4i 268 . 2  |-  ( y  e.  ran  ( A  o.  B )  <->  y  e.  ran  ( A  |`  ran  B
) )
1918eqriv 2313 1  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1532    = wceq 1633    e. wcel 1701   class class class wbr 4060   ran crn 4727    |` cres 4728    o. ccom 4730
This theorem is referenced by:  rnco2  5217  cofunexg  5780  1stcof  6189  2ndcof  6190  smobeth  8253  coeq0  25979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-xp 4732  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738
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