Theorem rnco 5117

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 2733	. . . . . 6 |- x e. _V
2		vex 2733	. . . . . 6 |- y e. _V
3	1, 2	brco 4782	. . . . 5 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
4	3	exbii 1598	. . . 4 |- ( E. x x ( A o. B ) y <-> E. x E. z ( x B z /\ z A y ) )
5		excom 1657	. . . 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 1895	. . . . . . 7 |- ( E. x ( x B z /\ z A y ) <-> ( E. x x B z /\ z A y ) )
8		vex 2733	. . . . . . . . 9 |- z e. _V
9	8	elrn 4854	. . . . . . . 8 |- ( z e. ran B <-> E. x x B z )
10	9	anbi2i 454	. . . . . . 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 4897	. . . . . 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 1598	. . . 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 4854	. . 3 |- ( y e. ran ( A o. B ) <-> E. x x ( A o. B ) y )
17	2	elrn 4854	. . 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 2167	1 |- ran ( A o. B ) = ran ( A |` ran B )

Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   class class class wbr 3989   ran crn 4612   |` cres 4613   o. ccom 4615

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623

This theorem is referenced by:  rnco2  5118  cofunexg  6088  1stcof  6142  2ndcof  6143  djudom  7070