Theorem rnco 5110

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.

Step	Hyp	Ref	Expression
1		vex 2729	. . . . . 6 |- x e. _V
2		vex 2729	. . . . . 6 |- y e. _V
3	1, 2	brco 4775	. . . . 5 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
4	3	exbii 1593	. . . 4 |- ( E. x x ( A o. B ) y <-> E. x E. z ( x B z /\ z A y ) )
5		excom 1652	. . . 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 1890	. . . . . . 7 |- ( E. x ( x B z /\ z A y ) <-> ( E. x x B z /\ z A y ) )
8		vex 2729	. . . . . . . . 9 |- z e. _V
9	8	elrn 4847	. . . . . . . 8 |- ( z e. ran B <-> E. x x B z )
10	9	anbi2i 453	. . . . . . 7 |- ( ( z A y /\ z e. ran B ) <-> ( z A y /\ E. x x B z ) )
11	6, 7, 10	3bitr4i 211	. . . . . 6 |- ( E. x ( x B z /\ z A y ) <-> ( z A y /\ z e. ran B ) )
12	2	brres 4890	. . . . . 6 |- ( z ( A |` ran B ) y <-> ( z A y /\ z e. ran B ) )
13	11, 12	bitr4i 186	. . . . 5 |- ( E. x ( x B z /\ z A y ) <-> z ( A |` ran B ) y )
14	13	exbii 1593	. . . 4 |- ( E. z E. x ( x B z /\ z A y ) <-> E. z z ( A |` ran B ) y )
15	4, 5, 14	3bitri 205	. . 3 |- ( E. x x ( A o. B ) y <-> E. z z ( A |` ran B ) y )
16	2	elrn 4847	. . 3 |- ( y e. ran ( A o. B ) <-> E. x x ( A o. B ) y )
17	2	elrn 4847	. . 3 |- ( y e. ran ( A |` ran B ) <-> E. z z ( A |` ran B ) y )
18	15, 16, 17	3bitr4i 211	. 2 |- ( y e. ran ( A o. B ) <-> y e. ran ( A |` ran B ) )
19	18	eqriv 2162	1 |- ran ( A o. B ) = ran ( A |` ran B )

Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   class class class wbr 3982   ran crn 4605   |` cres 4606   o. ccom 4608

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616

This theorem is referenced by:  rnco2  5111  cofunexg  6077  1stcof  6131  2ndcof  6132  djudom  7058