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Theorem rnco 5015
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 2663 . . . . . 6  |-  x  e. 
_V
2 vex 2663 . . . . . 6  |-  y  e. 
_V
31, 2brco 4680 . . . . 5  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
43exbii 1569 . . . 4  |-  ( E. x  x ( A  o.  B ) y  <->  E. x E. z ( x B z  /\  z A y ) )
5 excom 1627 . . . 4  |-  ( E. x E. z ( x B z  /\  z A y )  <->  E. z E. x ( x B z  /\  z A y ) )
6 ancom 264 . . . . . . 7  |-  ( ( E. x  x B z  /\  z A y )  <->  ( z A y  /\  E. x  x B z ) )
7 19.41v 1858 . . . . . . 7  |-  ( E. x ( x B z  /\  z A y )  <->  ( E. x  x B z  /\  z A y ) )
8 vex 2663 . . . . . . . . 9  |-  z  e. 
_V
98elrn 4752 . . . . . . . 8  |-  ( z  e.  ran  B  <->  E. x  x B z )
109anbi2i 452 . . . . . . 7  |-  ( ( z A y  /\  z  e.  ran  B )  <-> 
( z A y  /\  E. x  x B z ) )
116, 7, 103bitr4i 211 . . . . . 6  |-  ( E. x ( x B z  /\  z A y )  <->  ( z A y  /\  z  e.  ran  B ) )
122brres 4795 . . . . . 6  |-  ( z ( A  |`  ran  B
) y  <->  ( z A y  /\  z  e.  ran  B ) )
1311, 12bitr4i 186 . . . . 5  |-  ( E. x ( x B z  /\  z A y )  <->  z ( A  |`  ran  B ) y )
1413exbii 1569 . . . 4  |-  ( E. z E. x ( x B z  /\  z A y )  <->  E. z 
z ( A  |`  ran  B ) y )
154, 5, 143bitri 205 . . 3  |-  ( E. x  x ( A  o.  B ) y  <->  E. z  z ( A  |`  ran  B ) y )
162elrn 4752 . . 3  |-  ( y  e.  ran  ( A  o.  B )  <->  E. x  x ( A  o.  B ) y )
172elrn 4752 . . 3  |-  ( y  e.  ran  ( A  |`  ran  B )  <->  E. z 
z ( A  |`  ran  B ) y )
1815, 16, 173bitr4i 211 . 2  |-  ( y  e.  ran  ( A  o.  B )  <->  y  e.  ran  ( A  |`  ran  B
) )
1918eqriv 2114 1  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465   class class class wbr 3899   ran crn 4510    |` cres 4511    o. ccom 4513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521
This theorem is referenced by:  rnco2  5016  cofunexg  5977  1stcof  6029  2ndcof  6030  djudom  6946
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