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Theorem rnco 4827
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 2560 . . . . . 6  |-  x  e. 
_V
2 vex 2560 . . . . . 6  |-  y  e. 
_V
31, 2brco 4506 . . . . 5  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
43exbii 1496 . . . 4  |-  ( E. x  x ( A  o.  B ) y  <->  E. x E. z ( x B z  /\  z A y ) )
5 excom 1554 . . . 4  |-  ( E. x E. z ( x B z  /\  z A y )  <->  E. z E. x ( x B z  /\  z A y ) )
6 ancom 253 . . . . . . 7  |-  ( ( E. x  x B z  /\  z A y )  <->  ( z A y  /\  E. x  x B z ) )
7 19.41v 1782 . . . . . . 7  |-  ( E. x ( x B z  /\  z A y )  <->  ( E. x  x B z  /\  z A y ) )
8 vex 2560 . . . . . . . . 9  |-  z  e. 
_V
98elrn 4577 . . . . . . . 8  |-  ( z  e.  ran  B  <->  E. x  x B z )
109anbi2i 430 . . . . . . 7  |-  ( ( z A y  /\  z  e.  ran  B )  <-> 
( z A y  /\  E. x  x B z ) )
116, 7, 103bitr4i 201 . . . . . 6  |-  ( E. x ( x B z  /\  z A y )  <->  ( z A y  /\  z  e.  ran  B ) )
122brres 4618 . . . . . 6  |-  ( z ( A  |`  ran  B
) y  <->  ( z A y  /\  z  e.  ran  B ) )
1311, 12bitr4i 176 . . . . 5  |-  ( E. x ( x B z  /\  z A y )  <->  z ( A  |`  ran  B ) y )
1413exbii 1496 . . . 4  |-  ( E. z E. x ( x B z  /\  z A y )  <->  E. z 
z ( A  |`  ran  B ) y )
154, 5, 143bitri 195 . . 3  |-  ( E. x  x ( A  o.  B ) y  <->  E. z  z ( A  |`  ran  B ) y )
162elrn 4577 . . 3  |-  ( y  e.  ran  ( A  o.  B )  <->  E. x  x ( A  o.  B ) y )
172elrn 4577 . . 3  |-  ( y  e.  ran  ( A  |`  ran  B )  <->  E. z 
z ( A  |`  ran  B ) y )
1815, 16, 173bitr4i 201 . 2  |-  ( y  e.  ran  ( A  o.  B )  <->  y  e.  ran  ( A  |`  ran  B
) )
1918eqriv 2037 1  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   class class class wbr 3764   ran crn 4346    |` cres 4347    o. ccom 4349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357
This theorem is referenced by:  rnco2  4828  cofunexg  5738  1stcof  5790  2ndcof  5791
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