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Theorem rnco 5344
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 3050 . . . . . 6  |-  x  e. 
_V
2 vex 3050 . . . . . 6  |-  y  e. 
_V
31, 2brco 5008 . . . . 5  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
43exbii 1720 . . . 4  |-  ( E. x  x ( A  o.  B ) y  <->  E. x E. z ( x B z  /\  z A y ) )
5 excom 1929 . . . 4  |-  ( E. x E. z ( x B z  /\  z A y )  <->  E. z E. x ( x B z  /\  z A y ) )
6 ancom 452 . . . . . . 7  |-  ( ( E. x  x B z  /\  z A y )  <->  ( z A y  /\  E. x  x B z ) )
7 19.41v 1832 . . . . . . 7  |-  ( E. x ( x B z  /\  z A y )  <->  ( E. x  x B z  /\  z A y ) )
8 vex 3050 . . . . . . . . 9  |-  z  e. 
_V
98elrn 5078 . . . . . . . 8  |-  ( z  e.  ran  B  <->  E. x  x B z )
109anbi2i 701 . . . . . . 7  |-  ( ( z A y  /\  z  e.  ran  B )  <-> 
( z A y  /\  E. x  x B z ) )
116, 7, 103bitr4i 281 . . . . . 6  |-  ( E. x ( x B z  /\  z A y )  <->  ( z A y  /\  z  e.  ran  B ) )
122brres 5114 . . . . . 6  |-  ( z ( A  |`  ran  B
) y  <->  ( z A y  /\  z  e.  ran  B ) )
1311, 12bitr4i 256 . . . . 5  |-  ( E. x ( x B z  /\  z A y )  <->  z ( A  |`  ran  B ) y )
1413exbii 1720 . . . 4  |-  ( E. z E. x ( x B z  /\  z A y )  <->  E. z 
z ( A  |`  ran  B ) y )
154, 5, 143bitri 275 . . 3  |-  ( E. x  x ( A  o.  B ) y  <->  E. z  z ( A  |`  ran  B ) y )
162elrn 5078 . . 3  |-  ( y  e.  ran  ( A  o.  B )  <->  E. x  x ( A  o.  B ) y )
172elrn 5078 . . 3  |-  ( y  e.  ran  ( A  |`  ran  B )  <->  E. z 
z ( A  |`  ran  B ) y )
1815, 16, 173bitr4i 281 . 2  |-  ( y  e.  ran  ( A  o.  B )  <->  y  e.  ran  ( A  |`  ran  B
) )
1918eqriv 2450 1  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1446   E.wex 1665    e. wcel 1889   class class class wbr 4405   ran crn 4838    |` cres 4839    o. ccom 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-xp 4843  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849
This theorem is referenced by:  rnco2  5345  coeq0  5347  cofunexg  6762  1stcof  6826  2ndcof  6827  smobeth  9016  elmsubrn  30178  ftc1anclem3  32031
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