Theorem rnco 5243

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 2805	. . . . . 6 |- x e. _V
2		vex 2805	. . . . . 6 |- y e. _V
3	1, 2	brco 4901	. . . . 5 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
4	3	exbii 1653	. . . 4 |- ( E. x x ( A o. B ) y <-> E. x E. z ( x B z /\ z A y ) )
5		excom 1712	. . . 4 |- ( E. x E. z ( x B z /\ z A y ) <-> E. z E. x ( x B z /\ z A y ) )
6		ancom 266	. . . . . . 7 |- ( ( E. x x B z /\ z A y ) <-> ( z A y /\ E. x x B z ) )
7		19.41v 1951	. . . . . . 7 |- ( E. x ( x B z /\ z A y ) <-> ( E. x x B z /\ z A y ) )
8		vex 2805	. . . . . . . . 9 |- z e. _V
9	8	elrn 4975	. . . . . . . 8 |- ( z e. ran B <-> E. x x B z )
10	9	anbi2i 457	. . . . . . 7 |- ( ( z A y /\ z e. ran B ) <-> ( z A y /\ E. x x B z ) )
11	6, 7, 10	3bitr4i 212	. . . . . 6 |- ( E. x ( x B z /\ z A y ) <-> ( z A y /\ z e. ran B ) )
12	2	brres 5019	. . . . . 6 |- ( z ( A |` ran B ) y <-> ( z A y /\ z e. ran B ) )
13	11, 12	bitr4i 187	. . . . 5 |- ( E. x ( x B z /\ z A y ) <-> z ( A |` ran B ) y )
14	13	exbii 1653	. . . 4 |- ( E. z E. x ( x B z /\ z A y ) <-> E. z z ( A |` ran B ) y )
15	4, 5, 14	3bitri 206	. . 3 |- ( E. x x ( A o. B ) y <-> E. z z ( A |` ran B ) y )
16	2	elrn 4975	. . 3 |- ( y e. ran ( A o. B ) <-> E. x x ( A o. B ) y )
17	2	elrn 4975	. . 3 |- ( y e. ran ( A |` ran B ) <-> E. z z ( A |` ran B ) y )
18	15, 16, 17	3bitr4i 212	. 2 |- ( y e. ran ( A o. B ) <-> y e. ran ( A |` ran B ) )
19	18	eqriv 2228	1 |- ran ( A o. B ) = ran ( A |` ran B )

Syntax hints:   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   class class class wbr 4088   ran crn 4726   |` cres 4727   o. ccom 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737

This theorem is referenced by:  rnco2  5244  cofunexg  6270  1stcof  6325  2ndcof  6326  djudom  7291