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Theorem rnco 4855
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 2577 . . . . . 6  |-  x  e. 
_V
2 vex 2577 . . . . . 6  |-  y  e. 
_V
31, 2brco 4534 . . . . 5  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
43exbii 1512 . . . 4  |-  ( E. x  x ( A  o.  B ) y  <->  E. x E. z ( x B z  /\  z A y ) )
5 excom 1570 . . . 4  |-  ( E. x E. z ( x B z  /\  z A y )  <->  E. z E. x ( x B z  /\  z A y ) )
6 ancom 257 . . . . . . 7  |-  ( ( E. x  x B z  /\  z A y )  <->  ( z A y  /\  E. x  x B z ) )
7 19.41v 1798 . . . . . . 7  |-  ( E. x ( x B z  /\  z A y )  <->  ( E. x  x B z  /\  z A y ) )
8 vex 2577 . . . . . . . . 9  |-  z  e. 
_V
98elrn 4605 . . . . . . . 8  |-  ( z  e.  ran  B  <->  E. x  x B z )
109anbi2i 438 . . . . . . 7  |-  ( ( z A y  /\  z  e.  ran  B )  <-> 
( z A y  /\  E. x  x B z ) )
116, 7, 103bitr4i 205 . . . . . 6  |-  ( E. x ( x B z  /\  z A y )  <->  ( z A y  /\  z  e.  ran  B ) )
122brres 4646 . . . . . 6  |-  ( z ( A  |`  ran  B
) y  <->  ( z A y  /\  z  e.  ran  B ) )
1311, 12bitr4i 180 . . . . 5  |-  ( E. x ( x B z  /\  z A y )  <->  z ( A  |`  ran  B ) y )
1413exbii 1512 . . . 4  |-  ( E. z E. x ( x B z  /\  z A y )  <->  E. z 
z ( A  |`  ran  B ) y )
154, 5, 143bitri 199 . . 3  |-  ( E. x  x ( A  o.  B ) y  <->  E. z  z ( A  |`  ran  B ) y )
162elrn 4605 . . 3  |-  ( y  e.  ran  ( A  o.  B )  <->  E. x  x ( A  o.  B ) y )
172elrn 4605 . . 3  |-  ( y  e.  ran  ( A  |`  ran  B )  <->  E. z 
z ( A  |`  ran  B ) y )
1815, 16, 173bitr4i 205 . 2  |-  ( y  e.  ran  ( A  o.  B )  <->  y  e.  ran  ( A  |`  ran  B
) )
1918eqriv 2053 1  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409   class class class wbr 3792   ran crn 4374    |` cres 4375    o. ccom 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385
This theorem is referenced by:  rnco2  4856  cofunexg  5766  1stcof  5818  2ndcof  5819
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