Theorem rnco 5040

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 2684	. . . . . 6 |- x e. _V
2		vex 2684	. . . . . 6 |- y e. _V
3	1, 2	brco 4705	. . . . 5 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
4	3	exbii 1584	. . . 4 |- ( E. x x ( A o. B ) y <-> E. x E. z ( x B z /\ z A y ) )
5		excom 1642	. . . 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 1874	. . . . . . 7 |- ( E. x ( x B z /\ z A y ) <-> ( E. x x B z /\ z A y ) )
8		vex 2684	. . . . . . . . 9 |- z e. _V
9	8	elrn 4777	. . . . . . . 8 |- ( z e. ran B <-> E. x x B z )
10	9	anbi2i 452	. . . . . . 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 4820	. . . . . 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 1584	. . . 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 4777	. . 3 |- ( y e. ran ( A o. B ) <-> E. x x ( A o. B ) y )
17	2	elrn 4777	. . 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 2134	1 |- ran ( A o. B ) = ran ( A |` ran B )

Syntax hints:   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   class class class wbr 3924   ran crn 4535   |` cres 4536   o. ccom 4538

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546

This theorem is referenced by:  rnco2  5041  cofunexg  6002  1stcof  6054  2ndcof  6055  djudom  6971